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All Textbook Solutions for A Transition to Advanced Mathematics

Which of the following are propositions? Give the truth value of each proposition. What time is dinner? It is not the case that is not a rational number. x/2 is a rational number. 2x+3y is a real number. Either is rational and 17 is a prime, or 713 and 81 is a perfectsquare. Either 2 is rational and is irrational, or 2 is rational. Either 5 is rational and 4.9 is rational, or there are exactly four primesless than 10. 3.7 is rational, and either 310 or 315. It is not the case that 39 is prime, or that 64 is a power of 2. There are more than three false statements in this book, and this statementis one of them.For each pair of statements, determine whether the conjunction PQ and thedisjunction PQ are true. P is 2 and Q is “97 is a prime number.” P is “The moon is larger than Earth” and Q is “The prime divisors of 12are 2 and 3.” P is 52+122=132 and Q is 2+3=2+3. P is “France is south of Italy” and Q is “New Zealand is in Europe.” P is “0, 5, and 10 are all natural numbers” and Q is “98 has two primedivisors.” P is “Hexagons have 5 sides” and Q is 23=23.Make a truth table for each of the following propositional forms. P~P P~P P~Q P(Q~Q) (PP)~Q ~(PQ) (P~Q)R ~P~Q P(QR) (PQ)(PR) PP (PQ)(R~S)If P, Q, and R are true while S and K are false, which of the following are true? (a) Q(RS) (b) Q(RS) (c) (PQ)(RS) (d) (PQ)(RS) (e) PQ (f ) (QS)(QS) (g) (PS)(PK) (h) K(SQ)Use truth tables to verify each part of Theorem 1.1.1.Which of the following pairs of propositional forms are equivalent? (a) PQ,(PQ) (b) (P)(Q),(PQ) (c) (PQ)R,P(QR) (d) (PQ),PQ (e) (PQ)R,P(QR) (f) (PQ)P,P Determine the propositional form and truth value for each of the following: (a) It is not the case that gold is not a metal. (b) 19 and 79 are prime, but 119 is not. (c) Julius Caesar was born in 1492 or 1493 and died in 1776. (d) Perth or Panama City or Pisa is located in Europe. (e) Although 51 divides 153, it is neither prime nor a divisor of 409. (f) While the number is greater than 3, the sum 1+2 is less than 8. (g) It is not the case that both 5 and 13 are elements of , but 4 is in theset of rational numbers.Suppose P, Q, and R are propositional forms. Explain why each is true. If P is equivalent to Q, then Q is equivalent to P. If P is equivalent to Q, and Q is equivalent to R, then P is equivalent to R. If P is equivalent to Q, then ~P is equivalent to ~Q. If Q is equivalent to R, then PQ is equivalent to PR. If Q is equivalent to R, then PQ is equivalent to PR.Suppose P, Q, S, and R are propositional forms, P is equivalent to Q, and S isequivalent to R. For each pair of forms, determine whether they are necessarilyequivalent. If they are equivalent, explain why. (a) P and R (b) P and ~~Q (c) PS and QR (d) PS and QR (e) (PS) and QR ( f ) PQ and SR Use a truth table to determine whether each of the following is a tautology,a contradiction, or neither. (PQ)(PQ) (PP) (PQ)(PQ) (PQ)(PQ)(PQ)(PQ) (QP)(PR) P[(QP)(RQ)]Give a useful denial of each statement. Assume that each variable is somefixed object so that each statement is a proposition. x is a positive integer. Cleveland will win the first game or the second game. 53. 641,371 is a composite integer. Roses are red and violets are blue. K is not bounded or K is compact. M is odd and one-to-one. The matrix M is diagonal and invertible. The function g has a relative maximum at x=2 or x=4 and a relativeminimum at x=3. Neither zs nor zt is true. R is transitive but not symmetric.Restore parentheses to these abbreviated propositional forms. (a) PQS (b) QS(PQ) (c) PQPRPS (d) PQPQR Other logical connectives between two propositions P and Q are possible. 13. The word or is used in two different ways in English. We have presentedthe truth table for ?, the inclusive or, whose meaning is “one or the other orboth.” The exclusive or, meaning “one or the other but not both” and denoted has its uses in English, as in “She will marry Heckle or she will marryJeckle.” The “inclusive or” is much more useful in mathematics and is theaccepted meaning unless there is a statement to the contrary. (a) Make a truth table for the “exclusive or” connective V. (b) Show that AB is equivalent to (AB)(AB).Other logical connectives between two propositions P and Q are possible. 14. “NAND” and “NOR” circuits are commonly used as a basis for flash memorychips. A NAND B is defined to be the negation of “A and B.” A NOR B isdefined to be the negation of “A or B.” (a) Write truth tables for the NAND and NOR connectives. (b) Show that (ANANDB)(ANORB) is equivalent to (ANANDB). (c) Show that (ANANDB)(ANORB) is equivalent to (ANORB).Identify the antecedent and the consequent for each of the following conditional sentences. Assume that a, b, and f represent some fixed sequence, integer, or function, respectively. If squares have three sides, then triangles have four sides. If the moon is made of cheese, then 8 is an irrational number. b divides 3 only if b divides 9. The differentiability of f is sufficient for f to be continuous. A sequence a is bounded whenever a is convergent. A function fis bounded if f is integrable. 1+2=3 is necessary for 1+1=2. The fish bite only when the moon is full. A time of 3 minutes, 48 seconds or less is necessary to qualify for the Olympic team.2E What can be said about the truth value of Q when (a) P is false and PQ is true? (b) P is true and PQ is true? (c) P is true and PQ is false? (d) P is false and PQ is true? (e) P is true and PQ is false?Identify the antecedent and the consequent for each conditional sentence inthe following statements from this book. (a) Exercise 3 of Section 1.6 (b) Theorem 2.1.1(c) (c) The PMI, Section 2.4 (d) Theorem 3.3.1 (e) Theorem 4.7.2 (f) Corollary 5.3.6 Which of the following conditional sentences are true? (a) If triangles have three sides, then squares have four sides. (b) If hexagons have six sides, then the moon is made of cheese. (c) If 7+6=14 , then 5+5=10 . (d) The Nile River flows east only if 64 is a perfect square. (e) Earth has one moon only if the Amazon River flows into the North Sea. (f) If Euclid’s birthday was April 2, then rectangles have four sides. (g) 5 is prime if 2 is not irrational. (h) 1+1=2 is sufficient for 36 .Which of the following are true? Assume that x and y are fixed real numbers. (a) Triangles have three sides iff squares have four sides. (b) 7+5=12 if and only if 1+1=2 . (c) 5+6=6+5 iff 7+1=10 . (d) A parallelogram has three sides iff 27 is prime. (e) The Eiffel Tower is in Paris if and only if the chemical symbol forhelium is H. (f) 10+1311+12 iff 13121110 . (g) x20 if and only if x0 . (h) x2y2=0 iff (xy)(x+y)=0 . (i) x2+y2=50 if and only if (x+y)2=50 .Make truth tables for these propositional forms. (a) P(QP) . (b) (PQ)(QP) . (c) Q(QP) . (d) (PQ)(PQ) . (e) (PQ)(QR)PR . (f) [(QS)(QR)][(PQ)(SR)] .Prove Theorem 1.2.2 by constructing truth tables for each equivalence.Determine whether each statement qualifies as a definition. (a) y=f(x) is a linear function if its graph is a straight line. (b) y=f(x) is a quadratic function when it contains an x2 term. (c) A quadrilateral is a square when all its sides have equal length. (d) A triangle is a right triangle if the sum of two of its interior angles is 90 . (e) Two lines are parallel when their slopes are the same number. (f) A quadrilateral is a rectangle if all its interior angles are equal.10E Dictionaries indicate that the conditional meaning of unless is preferred, but there are other interpretations as a converse or a biconditional. Discuss thetranslation of each sentence. (a) I will go to the store unless it is raining. (b) The Dolphins will not make the playoffs unless the Bears lose all the rest of their games. (c) You cannot go to the game unless you do your homework first. (d) You won’t win the lottery unless you buy a ticket.Show that the following pairs of statements are equivalent. (PQ)R and R(PQ). (PQ)R and (PR)Q. P(QR) and (QR)P. P(QR) and (PR)Q. (PQ)R and (PQ)R. PQ and (PQ)(QP).13E Give, if possible, an example of a false conditional sentence for which (a) the converse is true. (b) the converse is false. (c) the contrapositive is false. (d) the contrapositive is true.Give the converse and contrapositive of each sentence of Exercises 10(a), (b),(f) and (g). Decide whether each converse and contrapositive is true or false.16E The inverse, or opposite, of the conditional sentence PQ is ~P~Q. Show that PQ and its inverse are not equivalent forms. For what values of the propositions P and Q are PQ and its inverse both true? Which is equivalent to the converse of a conditional sentence, the contrapositive of its inverse, or the inverse of its contrapositive?Translate the following English sentences into symbolic sentences with quantifiers. The universe for each is given in parentheses. Not all precious stones are beautiful. (All stones) All precious stones are not beautiful. (All stones) Some isosceles triangle is a right triangle. (All triangles) No right triangle is isosceles. (All triangles) Every triangle that is not isosceles is a right triangle. All people are honest or no one is honest. (All people) Some people are honest and some people are not honest. (All people) Every nonzero real number is positive or negative. (Real numbers) Every integer is greater than -4 or less than 6. (Real numbers) Every integer is greater than some integer. (Integers) No integer is greater than every other integer. (Integers) Between any integer and any larger integer, there is a real number. (Real numbers) There is a smallest positive integer. (Real numbers) No one loves everybody. (All people) Everybody loves someone. (All people) For every positive real number x, there is a unique real number y such that 2y=x. (Real numbers)For each of the propositions in Exercise 1, write a useful denial, and give a translation into ordinary English.Translate these definitions from the Appendix into quantified sentences. (a) The natural number a divides the natural number b. (b) The natural number n is prime. (c) The natural number n is composite. (d) The sets A and B are equal. (e) The set A is a subset of B. (f) The set A is not a subset of B.4E The sentence “People dislike taxes” might be interpreted to mean “All peopledislike all taxes,” “All people dislike some taxes,” “Some people dislike alltaxes,” or “Some people dislike some taxes.” Give a symbolic translation for each of these interpretations.Let T={17},U={6},V={24} , and W={2,3,7,26} . In which of these four different universes is the statement true? (a) (x) (x is odd x8 ). (b) (x) (x is odd x8 ). (c) (x) (x is odd x8 ). (d) (x) (x is odd x8 ).(a) Complete the following proof of Theorem 1.3.1(b). Proof: Let U be any universe. The sentence (x)A(x) is true in U iff . . . iff (x)A(x) is true in U. (b) Give a proof of part (b) of Theorem 1.3.1 that uses part (a) of that theorem.Which of the following are true? The universe for each statement is given in parentheses. (a) (x)(x+xx).() (b) (x)(x+xx).() (c) (x)(2x+3=6x+7).() (d) (x)(3x=x2).() (e) (x)(3x=x).() (f ) (x)(3(2x)=5+8(1x)).() (g) (x)(x2+6x+50).() (h) (x)(x2+4x+50).() (i) (x)(x2x+41isprime).() (j) (x)(x2x+41isprime).() (k) (x)(x3+17x2+6x+1000).() (l) (x)(y)[xy(w)(xwy)].()Give an English translation for each. The universe is given in parentheses. (x)(x1).() (!x)(x0x0).() (x) (x is prime x2x is odd). () (!x)(logex=1).() ~(x)(x20).() (!x)(x2=0).() (x) (x is odd x2 is odd). ()Which of the following are true in the universe of all real numbers? (a) (x)(y)(x+y=0) . (b) (x)(y)(x+y=0) . (c) (x)(y)(x2+y2=1) . (d) (x)[x0(y)(y0xy0)] . (e) (y)(x)(z)(xy=xz) . (f) (x)(y)(xy) . (g) (y)(x)(xy) . (h) (!y)(y0y+30) . (i) (!x)(y)(x=y2) . (j) (y)(!x)(x=y2) . (k) (!x)(!y)(w)(w2xy) .Let A(x) be an open sentence with variable x. (a) Prove Theorem 1.3.2 (a). (b) Show that the converse of Theorem 1.3.2 (a) is false. (c) Prove Theorem 1.3.2 (b). (d) Prove that (E!x)A(x) is equivalent to (x)[A(x)(y)(A(y)x=y)] . (e) Find a useful denial for (E!x)A(x) .Suppose the polynomials anxn+an1xn1+...+a0 and bnxn+bn1xn1+...+b0 are not equal. Which of the following must be true? (a) anbn . (b) aibi whenever 0in . (c) aibi for every isuch that 0in . (d) aibi for some isuch that 0in . (e) It is not the case that ai=bi for all isuch that 0in . (f )It is not the case that ai=bi for some isuch that 0in . (g) There is an isuch that 0in and aibi . (h) If ai=bi for all isuch that 0in1 , then anbn .Which of the following are denials of (!x)P(x) ? (a) (x)P(x)(x)P(x) . (b) (x)P(x)(y)(z)(yzP(y)P(z)) . (c) (x)[P(x)(y)(P(y)xy)] . (d) (x)(y)[(P(x)P(y))x=y] .Riddle: What is the English translation of the symbolic statement ?Analyze the logical form of each of the following statements and construct just the outline of a proof. Since the statements may contain terms with which you are not familiar, you should not (and perhaps could not) provide any details of the proof. Outline a direct proof that if (G,*) is a cyclic group, then (G,*) is abelian. Outline a direct proof that if B is a nonsingular matrix, then the determinant of B is not zero. Suppose A, B, and C are sets. Outline a direct proof that if A is a subset of B and B is a subset of C, then A is a subset of C. Outline a direct proof that if the maximum value of the differentiable function f on the closed interval [a,b] occurs at x0, then either x0=a or x0=b or f(x0)=0. Outline a direct proof that if A is a diagonal matrix, then A is invertible whenever all its diagonal entries are nonzero.A theorem of linear algebra states that if A andB are invertible matrices, thenthe product AB is invertible. As in Exercise 1, outline (a) a direct proof of the theorem. (b) a direct proof of the converse of the theorem.Verify that [(BM)L(ML)]B is a tautology. See the example on page 31.These facts have been established at a crime scene: (i) If Professor Plum is not guilty, then the crime took place in the kitchen. (ii) If the crime took place at midnight, then Professor Plum is guilty. (iii) Miss Scarlet is innocent if and only if the weapon was not the candlestick. (iv) Either the weapon was the candlestick or the crime took place in the library. (v) Either Miss Scarlet or Professor Plum is guilty. Use each of the following as a sixth clue to solve the case. Explain your answer. (a) The crime took place in the library. (b) The crime did not take place in the library. (c) The crime was committed at noon with the revolver. (d) The crime took place at midnight in the conservatory. (Give a complete answer.)5E Let a and b be real numbers. Prove that (a) ab|=|ab . (b) ab|=|ba . (c) |ab|=|a||b| ,for b0 . (d) a+b||a|+|b (The Triangle Inequality). (e) if ab , then bab . (f )if bab , then ab . (g) ||a||b|||ab| .Suppose a, b, c, and d are integers. Prove that (a) 2a1 is odd. (b) if a is even, then a+1 is odd. (c) if a is odd, then a+2 is odd. (d) a(a+1) is even. (e) 1 divides a. (f ) a divides a. (g) if a and b are positive and a divides b, then ab . (h) if a divides b, then a divides bc. (i) if a and b are positive andab =1, then a=b=1 . (j) if a and b are positive, a divides b, and b divides a, then a=b . (k) if a divides b and c divides d, then ac divides bd. (l) if ab divides c, then a divides c. (m) if ac divides bc, then a divides b.Give two proofs that if n is a natural number, then n2+n+3 is odd. (a) Use two cases. (b) Use Exercises 7(d) and 5(h).Let a, b, and c be integers and x, y, and z be real numbers. Use the technique of working backward from the desired conclusion to prove that (a) if x and y are nonnegative, then x+y2xy . Where in the proof do we use the fact that x and y are nonnegative? (b) if a divides b and a divides b+c , then a divides 3c. (c) if ab0 and bc0 , then ax2+bx+c=0 has two real solutions. (d) if x3+2x20 , then 2x+511 . (e) if an isosceles triangle has sides of length x, y, and z, where x=y and z=2xy , then it is a right triangle. (f) if x and y are positive and x2y225x29y2+2250 , then x3 and y5 or else x3 and y5 .Recall that except for degenerate cases, the graph of Ax2+Bxy+Cy2+Dx+Ey+F=0 is an ellipse iff B24AC0 , a parabola iff B24AC=0 , a hyperbola iff B24AC0 . Prove that whenever ACB0, the graph of the equation is an ellipse. Prove that the graph of the equation is a hyperbola if AC0 or BC4A0 . Prove that if the graph is a parabola, then BC=0 or A=B2/(4C) .Exercises throughout the text with this title ask you to examine “Proofs to Grade.” These are claims alleged to be true and supposed “proofs” of the claims. You should decide the merit of the claim and the validity of the proof, and then assign a grade of A (correct), if the claim and proof are correct, even if the proof is not the simplest or the proof you would have given. C (partially correct), if the claim is correct and the proof is largely correct. The proof may contain one or two incorrect statements or justifications, but the errors are easily correctable. F (failure), if the claim is incorrect, or the main idea of the proof is incorrect, or there are too many errors. You must justify assignments of grades other than A, and if the proof is incorrect, you must explain what is incorrect and why. (a) Suppose a is an integer. Claim. If a is odd, then a2+1 is even. “Proof.” Let a. Then, by squaring an odd we get an odd. An odd plus an odd is even. So a2+1 is even. (b) Suppose a, b, and c are integers. Claim. If a divides b and a divides c, then a divides b+c . “Proof.” Suppose a divides b and a divides c. Then for some integer q,b=aq , and for some integer q,c=aq . Then b+c=aq+aq=2aq=a(2q) , so a divides b+c . (c) Suppose x is a positive real number. Claim. The sum of x and its reciprocal is greater than or equal to 2. That is, x+1x2 . “Proof.” Multiplying by x, we get x2+12x . By algebra, x22x+10 . Thus, (x1)20 . Any real number squared is greater than or equal to zero, so x+1x2 is true. (d) Suppose m is an integer. Claim. If m2 is odd, then m is odd. “Proof.” Assume m is odd. Then m=2k+1 for some integer k. Therefore, m2=(2k+1)2=4k2+4k+1=2(2k2+2k)+1 , which is odd. Therefore, if m2 is odd, then m is odd. (e) Suppose a is an integer. Claim. a3+a2 is even. “Proof.” a3+a2=a2(a+1) , which is always an odd number times an even number. Therefore, a3+a2 is even.Analyze the logical form of each of the following statements, and construct just the outline of a proof by the given method. Do not provide any details of the proof. Outline a proof by contraposition that if (G,*) is a cyclic group, then (G,*) is abelian. Outline a proof by contraposition that if B is a nonsingular matrix, then the determinant of B is not zero. Outline a proof by contradiction that the set of natural numbers is not finite. Outline a proof by contradiction that if x is a nonzero real number, the multiplicative inverse of x is unique. Outline a two-part proof that the inverse of the function f from A to B is a function from B to A if and only if f is one-to-one and fis onto B. Outline a two-part proof that a subset A of the real numbers is compact if and only if A is closed and bounded.A theorem of linear algebra states that if A andB are invertible matrices, then the product AB is invertible. As in Exercise 1, (a) outline a proof of the theorem by contraposition. (b) outline a proof of the converse of the theorem by contraposition. (c) outline a proof of the theorem by contradiction. (d) outline a proof of the converse of the theorem by contradiction. (e) outline a two-part proof that A and B are invertible matrices if and only if the product AB is invertible.Let x, y, and z be integers. Write a proof by contraposition to show that (a) if x is even, then x+1 is odd. (b) if x is odd, then x+2 is odd. (c) if x2 is not divisible by 4, then x is odd. (d) if xy is even, then either x or y is even. (e) if x+y is even, then x and y have the same parity. (f ) if xy is odd, then both x and y are odd. (g) if 8 does not divide x21 , then x is even. (h) if x does not divide yz, then x does not divide z.Write a proof by contraposition to show that for any real number x, (a) if x2+2x0 , then x0 . (b) if x25x+60 , then 2x3 . (c) if x3+x0 , then x0 . (d) if (x+1)(x1)0 , then x1 . (e) if x(x4)3 , then x1 or x3 .A circle has center (2,4) . (a) Prove that (1,5) and (5,1) are not both on the circle. (b) Prove that if the radius of the circle is less than 5, then the circle does not intersect the line y=x6 . (c) Prove that if (0,3) is not inside the circle, then (3,1) is not inside thecircle.Suppose a and b are positive integers. Write a proof by contradiction to show that (a) if a divides b, then ab . (b) if ab is odd, then both a and b are odd. (c) if a is odd, then a+1 is even. (d) if ab is odd, then a+b is odd. (e) if ab and ab3 , then a=1 .7E 8E Prove by contradiction that if n is a natural number, then nn+1nn+2.Prove that 5 is not a rational number.Three real numbers, x, y, and z, are chosen between 0 and 1. Suppose that 0xyz1. Prove that at least two of the numbers x, y, and Z are within 12 unit from one another.Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. (a) Suppose m is an integer. Claim. If m2 is odd, then m is odd. “Proof.” Assume that m2 is not odd. Then m2 is even and m2=2k for some integer k. Thus 2k is a perfect square; that is, 2k is an integer. If 2k is odd, then 2k=2n+1 for some integer n, which means m2=2k=(2n+1)2=4n2+4n+1=2(2n2+2n)+1 . Thus m2 is odd, contrary to our assumption. Therefore, 2k=m must be even. Thus if m2 is not odd, then m is not odd. Hence if m2 is odd, then m is odd. (b) Suppose t is a real number. Claim. If t is irrational, then 5t is irrational. “Proof.” Suppose 5t is rational. Then 5t=p/q , where p and q are integers and q0 . Therefore, t=p/(5q) , where p and 5q are integers and 5q0 , so t is rational. Therefore, if t is irrational, then 5t is irrational. (c) Suppose x and y are integers. Claim. If x and y are even, then x+y is even. “Proof.” Suppose x and y are even but x+y is odd. Then, for some integer k, x+y=2k+1 . Therefore, x+y+(2)k=1 . The left side of the equation is even because it is the sum of even numbers. However, the right side, 1, is odd. Because an even cannot equal an odd, we have a contradiction. Therefore, x+y is even. (d) Suppose a, b, and c are integers. Claim. If a divides both b and c, then a divides b+c . “Proof.” Assume that a does not divide b+c . Then there is no integer k such that ak=b+c . However, a divides b, so am=b for some integer m; and a divides c, so an=c for some integer n. Thus am+an=a(m+n)=b+c . Therefore, k=m+n is an integer satisfying ak=b+c . Thus the assumption that a does not divide b+c is false, and a does divide b+c . (e) Suppose m and n are integers. Claim. If m2+n2 is even, then m and n have the same parity. “Proof.” Suppose m2+n2 is even, and m and n have opposite parity. We may assume that m is odd and n is even. Then for some integers j and k, m=2j+1 and n=2k , so m2+n2=4j2+4j+1+4k2=2(2j2+2j+2k2)+1 , which is odd. This is a contradiction. (f) Suppose a and b are positive integers. Claim. If a+1 divides b and b divides b+3 , then a=2 and b=3 . “Proof.” Assume a+1 divides b and b divides b=3 . Then b=(a+1)k and b+3=bj, for some integers k and j. Choose k=1 and j=2 . Then b+3=2b , so 2bb=3 . Thus b=3 . From b=(a+1)k , we have 3=(a+1)1 , so a=2 . Therefore, a=2 and b=3 .Prove that (a) there exist integers m and n such that 2m+7n=1 . (b) there exist integers m and n such that 15m+12n=3 . (c) there do not exist integers m and n such that 2m+4n=7 . (d) there do not exist integers m and n such that 12m+15n=1 . (e) for every integer t, if there exist integers m and n such that 15m+16n=t ,then there exist integers r and s such that 3r+8s=t . (f )if there exist integers m and n such that 12m+15n=1 , then m and n are both positive. (g) for every odd integer m, if m has the form 4k+1 for some integer k, then m+2 has the form 4j1 for some integer j. (h) for every integer m, if m is odd, then m2=8k+1 for some integer k. (i) for all odd integers m and n, if mn=4k1 for some integer k, then m or n is of the form 4j1 for some integer j.Prove that for all integers a, b, and c, If adivides b1 and a divides c1, then a divides bc1. if a divides b, then for all natural numbers n,an divides bn. if a is odd, c0, c divides a, and c divides a+2, then c=1. if there exist integers m and n such that am+bn=1 and c1, then c does not divide a or c does not divide b.Prove that if every even natural number greater than 2 is the sum of two primes, * then every odd natural number greater than 5 is the sum of three primes.Provide either a proof or a counterexample for each of these statements. For all positive integers x,x2+x+41 is a prime. (x)(y)(x+y=0). (Universe of all reals) (x)(y)(x1y0yxx). (Universe of all reals) For integers a, b, c, if a divides bc, then either a divides b or a divides c. For integers a, b, c, and d, if a divides bc and a divides cd, then a divides bd. For all positive real numbers x,x2x0. For all positive real numbers x,2xx+1. For every positive real number x, there is a positive real number y less than x with the property that for all positive real numbers z,yzz. For every positive real number x, there is a positive real number y with the property that if yx, then for all positive real numbers z,yzz.(a) Prove that the natural number x is prime if and only if x1 and there is no positive integer greater than 1 and less than or equal to x that divides x. (b)Prove that if p is a prime number and p3, then 3 divides p2+2. (Hint: When p is divided by 3, the remainder is 0, 1, or 2. That is, for some integer k,p=3k or p=3k+1 or p=3k+2.)Prove that (a) for every natural number n, 1n1 . (b) there is a natural number M such that for all natural numbers nM , 1n0.13 . (c) for every natural number n, there is a natural number M such that 2nM . (d) there is a natural number M such that for every natural number n, 1nM . (e) there is no largest natural number. (f ) there is no smallest positive real number. (g) For every integer k there exists an integer m such that for all natural numbers n, we have 0m+kn . (h) For every natural number n there is a real number r such that for allnatural numbers m and t, if tm1r , then t+n102 . (i) there is a natural number K such that 1r20.01 whenever r is a realnumber larger than K. (j) there exist integers L and G such that L < G and for every real number x, if LxG , then 4 0102x12 . (k) there exists an odd integer M such that for all real numbers rlarger thanM, we have 12r0.01 . (l) for every natural number x, there is an integer k such that 3.3x+k50 . (m) there exist integers x100 and y30 such that x+y128 and for allreal numbers r and s, if rx and sy , then (r50)(s20)390 . (n) for every pair of positive real numbers x and y where xy , there existsa natural number M such that if n is a natural number and nM , then 1n(yx) .Starting at 9 a.m. on Monday, a hiker walked at a steady pace from the trailhead up a mountain and reached the summit at exactly 3 p.m. The hikercamped for the night and then hiked back down the same trail, again starting at 9 a.m. On this second walk, the hiker walked very slowly for the first twohours, but walked faster on other parts of the trail and returned to the startingpoint in exactly six hours. Prove that there is some point on the trail that the hiker passed at exactly the same time on the two days.Show by example that each of the following deductions involving multiple quantifiers is not valid. Note that parts (b), (c), and (d) are the converses of thevalid deductions 3, 4, and 6, respectively, on page 59. (a) (x)P(x)(x)P(x) (b) (x)[P(x)Q(x)][(x)P(x)(x)Q(x)] (c) [(x)P(x)(x)Q(x)](x)[P(x)Q(x)] (d) (y)(x)P(x,y)(x)(y)P(x,y)Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. (a) Claim. Every polynomial of degree 3 with real coefficients has a real zero. “Proof.” The polynomial p(x)=x38 has degree 3, real coefficients, and a real zero (x=2) . Thus the statement “Every polynomial of degree 3 with real coefficients does not have a real zero” is false, and hence its denial, “Every polynomial of degree 3 with real coefficients has a real zero,” is true. (b) Claim. There is a unique polynomial whose first derivative is 2x+3 and which has a zero at x=1 . “Proof.” The antiderivative of 2x+3 is x2+3x+C . If we let p(x)=x2+3x4 , then p(x)=2x+3 and p(1)=0 . So p(x) is the desired polynomial. (c) Claim. Every prime number greater than 2 is odd. “Proof.” The prime numbers greater than 2 are 3,5,7,11,13,17,19, . None of these is even, so all of them are odd. (d) Claim. There exists an irrational number r such that r2 is rational. “Proof.” If 32 is rational, then r=3 s the desired example. Otherwise, 32 is irrational and ( 3 2 )2=(3)2=3 , which is rational. Therefore, either 3 or 32 is an irrational number r such that r2 is rational. (e) Claim. For every real number x, x0 . “Proof.” We proceed by three cases: x0 , x=0 , and x0 . Case 1. x0 . Choose, for example, x=4 . Then 4=4 . Thus x0 . Case 2. x=0 . Then 0=0 . Thus x0 . Case 3. x0 . Choose, for example, x=5 . Then 5=5 . Thus x0 . (f ) Claim. If x is prime, then x+7 is composite. “Proof.” Let x be a prime number. If x=2 , then x+7=9 , which is composite. If x2 , then x is odd, so x+7 is even and greater than 2. In this case, too, x+7 is composite. Therefore, if x is prime, then x+7 is composite. (g) Claim. If t is an irrational number, then t8 is irrational. “Proof.” Suppose there exists an irrational number t such that t8 is rational. Then t8=pq , where p and q are integers and q0 . Then t=pq+8=p+8qq , with p+8q and q integers and q0 . This is a contradiction because t is irrational. Therefore, for all irrational numbers t, t8 is irrational. (h) Claim. For real numbers x and y, if xy=0 , then x=0 or y=0 . “Proof.” Case 1. If x=0 , then xy=0y=0 . Case 2. If y=0 , then xy=x0=0 . In either case, xy=0 . (i) Claim. For every real number x in the interval (3,6) , there is a natural number K such that for every real number y, if yK , then 1y110 . “Proof.” Assume that x is in the interval (3,6) . Then x3 . Let K be x+7 . Then K10 . Suppose that y is a real number and yK . Then y10 , so 1y110 . ( j) Claim. For every natural number n, nn2 . “Proof.” Let n be a natural number. Since n is a natural number, 1n . Since n is positive, n1nn . Therefore, nn2 for all natural numbers n.(a) Let a be a negative real number. Prove that if a is a solution to the equation x2x6=0 , then a is a solution to x3+2x2+x+2=0 . (b) Let x be a real number. Prove that 0x3 implies x+1(x1)2 . (c) Let a and b be positive integers such that a0 . Prove that if a does not divide b, then there is no positive integer x that is a solution to ax2+bx+ba=0 . (d) Prove that if x is a real number and x1 , then 3|x2|x4 . (e) Prove that no point inside the circle (x3)2+y2=6 is on the line y=x+1 . (f) Let x be a real number. Prove that if x2+7x=9x+15 , then x2 or x4x30 . (g) Suppose that x is a real number. Prove that x2+7x100 if and only if x2 or x5 . (h) Prove that if two nonvertical lines are perpendicular, then the product of their slopes is -1. (Recall that nonvertical lines are those lines in the plane that have slope.) (i) Prove that there exists a point (x,y) inside the circle (x3)2+(y2)2=13 such that x0 and y3.99 . (j) Prove that there is a pair, and only one pair, (x,y) of real numbers such that 0x2,0y1 , and 3x2+2y214 .2E Prove that (a) 5n2+3n+4 is even, for all integers n. (b) for all integers n, if 5n+1 is even, then 2n2+3n+4 is odd. (c) the sum of five consecutive integers is always divisible by 5. (d) n3n is divisible by 6, for all integers n. (e) (n3n)(n+2) is divisible by 12, for all integers n. (f )every four-digit palindrome number is divisible by 11. (A palindrome number is a number that reads the same forward and backward.) (g) ifa, b, and c are real numbers and (abi)(cci)=1i , then ac=1 . (h) ifp is a prime integer, then p+19 is composite. (i) ifn is a natural number and r is the remainder when n is divided by 3, then if t=5 or 11, the sum r2+r+t is prime. (j) ifm and n are natural numbers, n4 , and (2n)(2m)2(mn) , then m=1 or n=3 . (k) if n2 is an even natural number, then 2n1 is not prime. (l) ifS is a set of real number such that a, b are in S, and if xa and xb for every element x of S, then a=b . (m) ifa and b be integers and b is odd, then 1 and -1 are not solutions of the equation ax4+bx2+a=0 . (n) if two nonvertical lines have slopes whose product is -1, then the lines are perpendicular.4E Prove that (a) if x + y is irrational, then either x or y is irrational. (b) if x is rational and y is irrational, then x+y is irrational. (c) there exist irrational numbers x and y such that x+y is rational. (d) for every rational number z, there exist irrational numbers x and y suchthat x+y=z . (e) for every rational number z and every irrational number x, there exists aunique irrational number y such that x+y=z . (f ) for every positive irrational number x, there is a positive irrational numbery such that y12 and yx .6E 7E 8E 9E 10E Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. (a) Claim. There is a unique three-digit number whose digits have sum 8 and product 10. “Proof.” Let x, y, and z be the digits. Then x+y+z=8 and xyz=10 . The only factors of 10 are 1, 2, 5, and 10, but since 10 is not a digit, the digits must be 1, 2, and 5. The sum of these digits is 8. Therefore, 125 is the only three-digit number whose digits have sum 8 and product 10. (b) Claim. There is a unique set of three consecutive odd numbers that are all prime. “Proof.” The consecutive odd numbers 3, 5, and 7 are all prime. Suppose that x, y, and z are consecutive odd numbers, all prime, and x3 . Then y=x+2 and z=x+4 . Since x is prime, when x is divided by 3, the remainder is 1 or 2. In case the remainder is 1, then x=3k+1 for some integer k1 . But then y=x+2=3k+3=3(k+1) , so y is not prime. In case the remainder is 2, then x=3k+2 for some integer k1 . But then z=x+4=3k+2+4=3(k+2) , so z is not prime. In either case we reach the contradiction that y or z is not prime. Thus x=3 and so y=5 and z=7 . Therefore, the only three consecutive odd primes are 3, 5, and 7. (c) Claim. If x is any real number, then either x is irrational or +x is irrational. “Proof.” It is known that p is an irrational number; that is, p cannot be written in the form ab for integers a and b. Consider x= . Then x=0 , which is rational, but +x=2 . If 2p were rational, then 2=ab for some integers a and b. Then =a2b , so p is rational. This is impossible, so 2p is irrational. Therefore, either x or +x is irrational. (d) Claim. If x is any real number, then either x is irrational or +x is irrational. “Proof.” It is known that p is an irrational number; that is, p cannot be written in the form ab for integers a and b. Let x be any real number. Suppose both x and +x are rational. Then, since the sum of two rational numbers is always rational, (x)+(+x)=2 is rational. Then 2=ab for some integers a and b. Then =a2b , so p is rational. This is impossible. Therefore, at least one of x or +x is irrational. (e) Claim. For all real numbers x and y, x23x=y23y2 if and only if x=y or x+y=3 . “Proof.” Suppose that x23x=y23y . Then x2+xy3xxyy2+3y=0 , so (x+y3)(xy)=0 . Therefore, x=y or x+y=3 . (f) Claim. For all real numbers x and y, the equality xy=12(x+y)2 holds if and only if x=y=0 . “Proof.” Part (i) Suppose that x=y=0 . Then xy=0=12(x+y)2 , so the equality holds. Part (ii) Suppose that x and y are real numbers and xy=12(x+y)2 . Then 2xy=x2+2xy+y2 , so x2+y2=0 . Since the square of a real number is never negative, x2=y2=0 , so x=y=0 . (g) Claim. If n is prime and n+5 or n+12 is prime, then n=2 . “Proof.” Assume that n is a prime number. Then n2 , so if n+5 is prime, then n+5 must be odd. Therefore, n must be even. Since 2 is the only even prime, n=2 . (h) Claim. Let a, b, and c be real numbers with a0 . If ax2+bx+c=0 has no rational roots, then cx2+bx+a=0 has no rational roots. “Proof.” Suppose that cx2+bx+a=0 has a rational root p/q. Then c(p/q)2+b(p/q)+a=0 . Then c+b(q/p)+a(q/p)2=0 , so q/p is a rational root of the equation ax2+bx+c=0 .For each given pair a, b of integers, find the unique quotient and remainderwhen b is divided by a. (a) a=8,b=310 (b) a=5,b=36 (c) a=5,b=36 (d) a=5,b=36 (e) a=7,b=44 (f ) a=8,b=52 (g) a=8,b=52 2E Let a and b be integers, a0 , and ab . Prove that when b is divided by a, the quotient is 0 if and only if b0 .4E 5E 6E 7E 8E Prove that for every prime p and for all natural numbers a, (a) g cd(p,a)=p iff p divides a. (b) gcd(p,a)=1 iff p does not divide a.Let q be a natural number greater than 1 with the property that q divides a orq divides b whenever q divides ab. Prove that q is prime.11E 12E Let a and b be nonzero integers that are relatively prime, and let c be an integer. Prove that the equation ax+by=c has an integer solution.Let a and b be nonzero integers and d=gcd(a,b) . Let m=bd and n=ad . Showthat if x=s and y=t is a solution to ax+by=c , then so is x=s+km and y=tkn for every integer k. (This shows how linear combinations help describe solutions to equations.)Let a and b be nonzero integers and c be an integer. Prove that the equation ax+by=c has an integer solution if and only if gcd(a,b) divides c.16E 17E Let a and b be integers, and let m=lcm(a,b) . Use the Division Algorithm toprove that if c is a common multiple of a and b, then m divides c.The greatest common divisor of positive integers a and b may be calculated using prime factorizations. Suppose a=p1r1p2r2pkrk and b=q1s1q2s2qmsm are the prime factorizations. Then gcd(a,b)=w1t1w2t2wntn, where each wi is a prime factor of both a and b and whenever wi=pu=qv then the exponent ti is the smaller of the corresponding exponents ru and sv. Use this method to find the gcd of each of the following pairs. a=84,b=30 a=132,b=42 a=360,b=540 a=315,b=180 20E 21E The Cayley tables for operations o,*,+, and are listed below. Which of the operations are commutative? Which of the operations are associative? Which systems have an identity? What is the identity element? For those systems that have an identity, which elements have inverses?Let m,n and M=A:A is an mn matrix with real number entries}. Let be matrix multiplication. Under what conditions on m and n is (M,) an algebraic system? Is the operation commutative? Explain. Let + be matrix addition. Under what conditions on m and n is (M,+) an algebraic system? Is the operation commutative? Explain.Let be an associative operation on nonempty set A with identity e. Suppose that a, b, c, and d are elements of A, b is the inverse of a, and d is the inverse of c. Prove that db is the inverse of ac.Let be an associative operation on nonempty set A with identity e. Suppose that a, b, c, and d are elements of A, b is the inverse of a, and d is the inverse of c. Prove that db is the inverse of ac.Suppose that (A,*) is an algebraic system and * is associative on A. Prove that if a1,a2,a3, and a4 are in A, then (a1*a2)*(a3*a4)=a1*((a2*a3)*a4). Use complete induction to prove that any product of n elements a1,a2,a3,...,an in that order is equal to the left-associated product (...(( a 1 * a 2 )*a3)..)*an. Thus, the product of n elements is alwaysthe same, no matter how they are grouped by parentheses, as long as theorder of the factors is not changed.Let (A,o) be an algebra structure. An element lA is a left identity for o if la=a for every aA. Give an example of a 3-element structure with exactly two left identities. Define a right identity for (A,o). Prove that if (A,o) has a right identity r and a left identity l, then r=l, and that r=l is an identity for o.Let G be a group. Prove that if a2=e for all aG, then G is abelian.Give an example of an algebraic structure of order 4 that has both right and left cancellation but that is not a group.9E Construct the operation table for each of the following: (8,+),(8,), and (U8,) (5,+),(5,), and (U5,) (10,+),(10,), and (U10,) (11,+),(11,), and (U11,)11E 12E Suppose m and m2. Prove that 1 and m1 are distinct units in (m,).Let m and a be natural numbers with am. Complete the proof of Theorem 6.1.3 by proving that if a is a unit in (m,) then a and m are relatively prime. if a is a divisor of zero in (Zm,), then a and m are not relatively prime.Complete the proof of Theorem 6.1.4. First, show that 1 is an element of (Um,).16E 17E 18E Repeat Exercise 2 with the operation * given by the table on the right.1E Let G be a group and aiG for all n. Prove that (a1a2a3)1=a31a21a11. State and prove a result similar to part (a) for n elements of G, for all n.Prove part (d) of Theorem 6.2.3. That is, prove that if G is a group, a, b, and c are elements of G, and ca=cb, then a=b.Prove part (b) of Theorem 6.2.4.List all generators of each cyclic group in Exercise 9.Let G be a group with identity e. Let aG. Prove that the set Ca={xG:xa=ax}, called the centralizer of a in G, is a subgroup of G. Let C=xG: for all yG,xy=yx. Prove that C, the center of G, is a subgroup of G. Let aG. Prove that the center of G is a subgroup of the centralizer of a in G.Let G be a group, and let H be a subgroup of G. Let a be a fixed element of G. Prove that K={a1ha:hH} is a subgroup of G.Let ({0},) be the group of nonzero complex numbers with complex number multiplication. Let =1+i32. (a) Find a. (b) Find a generator of a other than .Prove that for every natural number m greater than 1, the group (m,+) is cyclic with generators 1 and m1.Show that the structure ({1},), with operation defined by ab=a+bab, is an abelian group. You should first show that ({1},) is an algebraic structure.(a)In the group G of Exercise 2, find x such that vx=e;x such that vx=u;x such that vx=v; and x such that vx=w. (b)Let (G,*) be a group and a,bG. Show that there exist unique elements x and y in G such that a*x=b and y*a=b.Show that (,), with operation # defined by ab=a+b+1, is a group. Find x such that 50x=100.13E 14E 15E Show that each of the following algebraic structures is a group. Which groups are abelian? ({1,1},), where is integer multiplication. ({1,,},), where =1+i32,=1i32, and is complexnumber multiplication., ({1,1,i,i},), where is complex number multiplication. (P(X),), where X is a nonempty set and is the symmetric difference operation AB=(AB)(BA). (t,+), where t is a natural number. The set of 22 real matrices with determinant 1, where the operation is matrix multiplication.17E Given that G={e,u,v,w} is a group of order 4 with identity e and u2=v2=w2=e, construct the operation table for G.Give an example of an algebraic system (G,o) that is not a group such that in the operation table for o, every element of G appears exactly once in every row and once in every column. This can be done with as few as three elements in G.(a)What is the order of S4, the symmetric group on four elements? (b)Compute these products in S4:[1243][4213],[4321][4321], and [2143][1324]. (c)Compute these products in S4:[3124][3214],[4321][3124], and [1432][1432]. (d)Find the inverses of [1342],[4123], and [2143]. (d)Show that S4 is not abelian.Find the order of the element 3 in each group. (4,+) (5,+) (6,+) (8,+) (9,+) (U5,) (U7,) (U11,)Find the order of each element of the group S3. (7,+). (8,+). (U11,).Let 3 and 6 be the sets of integer multiples of 3 and 6, respectively. Let f be the function from (3,+) to (6,+) given by f(x)=4x. Prove that fis a homomorphism. Prove that fis one-to-one. What group is the homomorphic image of (3,+) under f?Let (3,+) and (6,+) be the groups in Exercise 10, and let g be the function from 3 to 6 given by g(x)=x+3. Is g a homomorphism? Explain.Let ({a,b,c},o) be the group with the operation table shown here. Verify that the mapping g:(6,+)({a,b,c},o) defined by g(0)=g(3)=a,g(1)=g(4)=b, and g(2)=g(5)=c is a homomorphism.(a)Prove that the function f:1824 given by f(x)=4x is well defined and is a homomorphism from (18,+) to (24,+). (b)FindRng (f), and give the operation table for the subgroup Rng (f) of 24.Define f:1512 by f(x)=4x. Prove that f is a well-defined function and a homomorphism from (15,+) to (12,+). Find Rng (f), and give the operation table for this subgroup of 12.Let (G,) and (H,*) be groups, i be the identity element for H, and h:(G,o)(H,*) be a homomorphism. The kernel of f is ker(f)={xG:f(x)=i}. Thus, ker(f) is all the elements of G that map to the identity in H. Show that ker(f) is a subgroup of G.Show that (4,+) and ({1,1,i,i},) are isomorphic.Prove that every subgroup of a cyclic group is cyclic.Let G=a be a cyclic group of order 30. What is the order of a6? List all elements of order 2. List all elements of order 3. List all elements of order 10.Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. If H and K are subgroups of a group G, then H n K is a subgroup of G. "Proof." Let a,bHK. Then a,bH and a,bK. Because H and Kare subgroups, ab1H and ab1K. Therefore, ab1HK. Claim. If H is a subgroup of a group G and xH, then xH={xh:hH} is a subgroup of G. "Proof." First, the identity eH. Thus, x=xexH. Therefore, xH. Second, let a,bxH. Then a=xh and b=xk for some h,kH. Then we have ab1=(xh)(xk)1=(xh)(k1)(x1)=x(hk1x1)xH. Therefore, xH is a subgroup of G.Find all subgroups of (8,+). (U11,). (5,+). (U7,). (J,*) with J={a,b,c,d,e,f} and the table for * shown at the right.In the group S4, find two different subgroups that have three elements. find two different subgroups that have four elements. [2314][3124]=[1234]. Is there a subgroup of S4 that contains [2314] but not [3124]? Explain. find the smallest subgroup that contains [4213] and [3241]. find the smallest subgroup that contains [2314] and [3421].Prove that if G is a group and H is a subgroup of G, then the inverse of an element xH is the same as its inverse in G (Theorem 6.3.1 (b)).(a)Prove that if H and K are subgroups of a group G, then HK is a sub- group of G. (b)Prove that if {H:} is a family of subgroups of a group G, then H is a subgroup of G. (c)Give an example of a group G and subgroups H and K of G such that HK is not a subgroup of G.Let G be a group and H be a subgroup of G. If H is abelian, must G be abelian? Explain.Prove or disprove: Every abelian group is cyclic.Let G be a group. If H is a subgroup of G and K is a subgroup of H, prove that K is a subgroup of G.Define f:++ by f(x)=x where + is the set of all positive real numbers. Is f:(+,+)(+,+) operation preserving? Is f:(+,)(+,) operation preserving?Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. Let o be the operation on defined by setting (a,b)(c,d)=(a+c,b+d), and let - be the usual subtraction on . Then the function f given by f(a,b)=a3b is an OP map from (,) to (,). "Proof." (4,2) and (3,1) are in . Then f((4,2)(3,1))=f(7,3)=733=2, whereas f(4,2)f(3,1)=20=2, sof is operation preserving. Claim. Let f:(G,*)(H,) and g:(H,)(K,) be OP maps. Then the composite gf: (G,*)(K,) is an OP map. "Proof." gf(ab)=g(f(ab))=g(f(a)f(b))=g(f(a))g(f(b))=(gf(a))(gf(b)).Define f: by f(x)=x3. Is f:(,+)(,+) operation preserving? Is f:(,)(,) operating preserving?Define on by setting (a,b)(c,d)=(acbd,ad+bc). Show that (,) is an algebraic system. Show that the function h from the system (,) to (,) given by h(a+bi)=(a,b) is a one-to-one function from the set of complex numbers that is onto and is operation preserving.Let f the set of all real-valued integrable functions defined on the interval [a,b]. Then (F,+) is an algebraic structure, where + is the addition of functions. Define I:(F,+)(,+) by I(f)=abf(x)dx. Use your knowledge of calculus to verify that I is an OP map.6E Let M be the set of all 22 matrices with real entries. Define M by Det[abcd]=adbc. Prove that Det:(M,)(,) is operation preserving, where (M,) denotes M with matrix multiplication. Prove that Det:(M,+)(,+) is not operation preserving, where (M,+) denotes M with matrix addition.Let Conj: be the conjugate mapping for complex numbers given by Conj(a+bi)=abi. Prove that Conj:(,+)(,+) is operation preserving, where (,+) denotes the complex numbers with addition. Prove that Conj:(,)(,) is operation preserving, where (,) denotes the complex numbers with multiplication.Prove the remaining parts of Theorem 6.4.1.Is S3 isomorphic to (6,+)? Explain.11E Use the method of proof of Cayley's Theorem to find a group of permutations isomorphic to (3,+). (5,+). (,+).Let (R,+,) be an algebraic structure such that (R,+) is an abelian group with identity 0. (R{0},) is an abelian group with identity 1. For all a, b, cR,a(b+c)=(ab)+(ac). 01. Prove that (R,+,) is a field by showing that for all a, b, cR,(a+b)c=(ac)+(bc). R has no divisors of zero.Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. If (R,+,) is a ring, a,bR, and a0, then the equation ax=b has a unique solution. "Proof." Suppose that p and q are two solutions to ax=b. Then ap=b and aq=b. Therefore, ap=aq. Therefore, p=q. Claim. If (R,+,) is a finite integral domain, then (R,+,) is a field. "Proof." Suppose that R has n elements. Let xR . Then the n+1 powers of x:e=x0,x,x2,x3,...,xn are not all distinct. Therefore, xt=xr for integers t and r, where we may assume that tr. Then xtxt=xtxr, Therefore, e=xrt. and therefore e=xxrt1. Thus, e=xxrt1. has an inverse. Hence R is a field. Claim. Let (R,+,) be an integral domain with a, b, cR and unity element 1. If ab=ac and a0, than b=c. "Proof." Suppose that ab=ac and a0. Then b=1b=(a1a)b=a1(ab)=a1(ac)=(a1a)c=1c=c. Therefore b=c.Which of the following is a ring with the usual operations of addition and multiplication? For each structure that is not a ring, list the ring axioms that are not satisfied. the closed interval [1,1] {a+bi:a,b}, where i2=1 {bi:b}, where i2=1 Let [2] be the set {a+b2:a,b}. Define addition and multiplication on [2] in the usual way. That is, (a+b2)+(c+d2)=(a+c)+(b+d)2 and (a+b2)(c+d2)=ac+ad2+bc2+bd(2)2 =ac+2bd+(ad+bc)2. Prove that ([2],+,) is a ring.Complete the proof that for every m,(m+,) is a ring (Theorem 6.5.1) by showing that (b+c)a=ba+ca for all a, b, and c in m.Define addition and multiplication on the set as follows. For a,b,c,d,(a,b)(c,d)=(a+c,b+d) and (a,b)(c,d)=(ac,bd). Prove that (,,) is a ring.7E Let (R,+,) be a ring and a,b,R. Prove that b+(a) is the unique solution to the equation x+a=b.Prove the remaining parts of Theorem 6.5.3: For all a,b,c, a0=0. (a)b=(ab). (ab)c=(ac)(bc).10E 11E 12E 13E 14E 1E Let A and B be subsets of . Prove that if sup(A) and sup(B) exist, then sup(AB) exists, and sup(AB)=max{sup(A),sup(B)}. State and prove a similar result for inf(AB).(a)Give an example of sets A and B of real numbers such that AB,sup(AB)sup(A), and sup(AB)sup(B). (b)For sets A and B such that AB, state and prove a relationship among sup(A), sup(B), and sup(AB).(a)Give an example of sets A and B of real numbers such that AB,inf(AB)inf(A), and inf(AB)inf(B). (b)For sets A and B such that AB, state and prove a relationship among inf(A), inf(B), and inf(AB).5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E Use the definition of “divides” to explain (a) why 5 divides 65 and (b) why 7does not divide 23.18E 19E 20E For each function, find the value of f at 3 and the image of 5. If possible, findpre-images of 0 and 2. Then give the domain and range of the function. f(x)=2x+6 f(x)=2x22 f(x)=1/(x4) f(x)=25x Let A be the set {1,2,3,4} and B={0,1,2,3}. Give a rule of correspondencefrom A to B that is not a function. a function with range B a function with range {0,1}.Formulate and prove a characterization of greatest lower bounds similar to that in Theorem 7.1.2 for least upper bounds.24E 25E Let 3 and 6 be the sets of integer multiples of 3 and 6, respectively. Let f be the function from (3,+) to (6,+) given by f(x)=4x. Prove that fis a homomorphism. Prove that fis one-to-one. What group is the homomorphic image of (3,+) under f?Let (3,+) and (6,+) be the groups in Exercise 10, and let g be the function from 3 to 6 given by g(x)=x+3. Is g a homomorphism? Explain.Let ({a,b,c},o) be the group with the operation table shown here. Verify that the mapping g:(6,+)({a,b,c},o) defined by g(0)=g(3)=a,g(1)=g(4)=b, and g(2)=g(5)=c is a homomorphism.(a)Prove that the function f:1824 given by f(x)=4x is well defined and is a homomorphism from (18,+) to (24,+). (b)FindRng (f), and give the operation table for the subgroup Rng (f) of 24.Define f:1512 by f(x)=4x. Prove that f is a well-defined function and a homomorphism from (15,+) to (12,+). Find Rng (f), and give the operation table for this subgroup of 12.Let (G,) and (H,*) be groups, i be the identity element for H, and h:(G,o)(H,*) be a homomorphism. The kernel of f is ker(f)={xG:f(x)=i}. Thus, ker(f) is all the elements of G that map to the identity in H. Show that ker(f) is a subgroup of G.7E 8E Prove that the relation of isomorphism is an equivalence relation. That is, prove that if (G,) is a group, then (G,) is isomorphic to (G,). if (G,) is isomorphic to (H,*) then (H,*) is isomorphic to (G,). if (G,) is isomorphic to (H,*) and (H,*) is isomorphic to (K,), then (G,) is isomorphic to (K,).10E Prove that if G is a group and H is a subgroup of G, then the inverse of an element xH is the same as its inverse in G (Theorem 6.3.1 (b)).12E 13E 14E 15E 16E 17E (a)Show that any two groups of order 2 are isomorphic. (b)Show that any two groups of order 3 are isomorphic. (c)Prove that there exist two groups of order 4 that are not isomorphic.(a)Show that the function h: defined by h(x)=3x is not a ring homomorphism. (b)Show that the function h:66 defined by h(x)=2x is not a ring homomorphism. (c)Let [x] be the set of all polynomials p(x) in the variable x with integer coefficients. Show that the function g:[x] defined by g(p(x))=p(0) is a ring homomorphism.Let R be the equivalence relation on ({0}) given by (x,y)R(u,v) if xv,yu. Let P be the set of equivalence classes of ({0}) modulo R. For (a,b) and (c,d) in P, define the operations and by (a,b)(c,d)=(ad+bc,bd) and (a,b)(c,d)=(ac,bd). Find the additive identity in this ring, the unity element, and the additive and multiplicative inverses of (2,5). Hint: For each answer, you must give a representative of the equivalence class in P. Suppose that f:(,+,)(P,,) is given by f(p/q)=(p,q). Prove that fis a ring homomorphism.Let (R,+,) be an integral domain. Prove that 0 has no multiplicative inverse.Complete the proof of Theorem 6.5.5. That is, prove that if (R,+,) is an integral domain, a,b,cR, and a0, then ba=ca implies b=c.6E Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. If (R,+,) is a ring, a,bR, and a0, then the equation ax=b has a unique solution. "Proof." Suppose that p and q are two solutions to ax=b. Then ap=b and aq=b. Therefore, ap=aq. Therefore, p=q. Claim. If (R,+,) is a finite integral domain, then (R,+,) is a field. "Proof." Suppose that R has n elements. Let xR . Then the n+1 powers of x:e=x0,x,x2,x3,...,xn are not all distinct. Therefore, xt=xr for integers t and r, where we may assume that tr. Then xtxt=xtxr, Therefore, e=xrt. and therefore e=xxrt1. Thus, e=xxrt1. has an inverse. Hence R is a field. Claim. Let (R,+,) be an integral domain with a, b, cR and unity element 1. If ab=ac and a0, than b=c. "Proof." Suppose that ab=ac and a0. Then b=1b=(a1a)b=a1(ab)=a1(ac)=(a1a)c=1c=c. Therefore b=c.8E 9E Use the method of proof of Cayley's Theorem to find a group of permutations isomorphic to (3,+). (5,+). (,+).11E Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. Let o be the operation on defined by setting (a,b)(c,d)=(a+c,b+d), and let - be the usual subtraction on . Then the function f given by f(a,b)=a3b is an OP map from (,) to (,). "Proof." (4,2) and (3,1) are in . Then f((4,2)(3,1))=f(7,3)=733=2, whereas f(4,2)f(3,1)=20=2, sof is operation preserving. Claim. Let f:(G,*)(H,) and g:(H,)(K,) be OP maps. Then the composite gf: (G,*)(K,) is an OP map. "Proof." gf(ab)=g(f(ab))=g(f(a)f(b))=g(f(a))g(f(b))=(gf(a))(gf(b)).13E Define on by setting (a,b)(c,d)=(acbd,ad+bc). Show that (,) is an algebraic system. Show that the function h from the system (,) to (,) given by h(a+bi)=(a,b) is a one-to-one function from the set of complex numbers that is onto and is operation preserving.15E Let f:(A,)(B,*) and g:(B,*)(C,X) be OP maps. Prove that gf is an OP map. Prove that if f1 is a function, then f1 is an OP map.17E Let Conj: be the conjugate mapping for complex numbers given by Conj(a+bi)=abi. Prove that Conj:(,+)(,+) is operation preserving, where (,+) denotes the complex numbers with addition. Prove that Conj:(,)(,) is operation preserving, where (,) denotes the complex numbers with multiplication.Prove the remaining parts of Theorem 6.4.1.Let 3={3k:k}. Apply the Subring Test (Exercise 8(b)) to show that (3+,) is a subring of (,+,).Use these exercises to check your understanding. Answers appear at the end of the Answers to Selected Exercises. 1. Write each set in two ways: by listing its elements and by stating the propertythat determines membership in the set. (a) The set of integers between 6 and 12 (b) The set of integers whose square is less than 17 (c) The set of solutions to x281=0 (d) The set of integer powers of 2 (e) The set of ingredients in a peanut butter and jelly sandwich Use these exercises to check your understanding. Answers appear at the end of the Answers to Selected Exercises. 2. What is the set of all real numbers x such that (a) x2=4? (b) x2=4?Use these exercises to check your understanding. Answers appear at the end of the Answers to Selected Exercises. 3. Which of these sets are finite? (a) The set of grains of sand on Earth (b) The set of integer powers of 2 (c) The set of three-letter words in English (d) The set of solutions to the equation 3x175x2437=0 (e) The set of real numbers between 0 and 1 Use these exercises to check your understanding. Answers appear at the end of the Answers to Selected Exercises. 4. (a) List all eight subsets of the set A={3,5,7}. (b) Let A={j,m,h} . Explain why {A} is not a subset of A.6E Use the definition of “divides” to explain (a) why 5 divides 65 and (b) why 7does not divide 23.8E 9E Complete the proof that for every m,(m+,) is a ring (Theorem 6.5.1) by showing that (b+c)a=ba+ca for all a, b, and c in m.Define addition and multiplication on the set as follows. For a,b,c,d,(a,b)(c,d)=(a+c,b+d) and (a,b)(c,d)=(ac,bd). Prove that (,,) is a ring.12E Let (R,+,) be a ring and a,b,R. Prove that b+(a) is the unique solution to the equation x+a=b.Prove the remaining parts of Theorem 6.5.3: For all a,b,c, a0=0. (a)b=(ab). (ab)c=(ac)(bc).We define a subring of a ring in the same way we defined a subgroup of a group: (S,+,) is a subring of the ring (R,+,) if SR, and (S,+,) is a ring with the same operations. For example, the ring of even integers is a subring of the ring of integers, and both are subrings of the ring of rational numbers. Prove that the ring ({0},+,) is a subring of any ring (R,+,) (called the trivial subring). (Subring Test) Prove that if (R,+,) is a ring, Tis a nonempty subset of R, and T is closed under subtraction and multiplication, then (T,+,) is a subring.1E 2E If possible, give an example of a set A such that sup(A)=4 and 4A. a set A such that sup(A)=4 and 4A. a set A such that sup(A)=4 and 4A. a set A such that sup(A)4 and 4A.Let A. Prove that if sup(A) exists, then sup(A)=infu:u is an upper bound of A}. if inf(A) exists, then inf(A)=supl:l is a lower bound of A}.Let A and B be subsets of . Prove that if sup(A) and sup(B) exist, then sup(AB) exists, and sup(AB)=max{sup(A),sup(B)}. State and prove a similar result for inf(AB).a.Give an example of sets A and B of real numbers such that AB,sup(AB)sup(A), and sup(AB)sup(B). b.For sets A and B such that AB, state and prove a relationship among sup(A), sup(B), and sup(AB).a.Give an example of sets A and B of real numbers such that AB,inf(AB)inf(A), and inf(AB)inf(B). b.For sets A and B such that AB, state and prove a relationship among inf(A), inf(B), and inf(AB).An alternate version of the Archimedean Principle for the reals has the effect of saying that there are no infinitesimal (infinitely small) real numbers. It says (0)(n)(1n). Prove that the two versions are equivalent.9E 10E 11E 12E 1E 2E Let A be a subset of . Prove that the set of all interior points of A is an open set.4E Let be an associative operation on nonempty set A with identity e. Suppose that a, b, c, and d are elements of A, b is the inverse of a, and d is the inverse of c. Prove that db is the inverse of ac.Suppose that (A,*) is an algebraic system and * is associative on A. Prove that if a1,a2,a3, and a4 are in A, then (a1*a2)*(a3*a4)=a1*((a2*a3)*a4). Use complete induction to prove that any product of n elements a1,a2,a3,...,an in that order is equal to the left-associated product (...(( a 1 * a 2 )*a3)..)*an. Thus, the product of n elements is alwaysthe same, no matter how they are grouped by parentheses, as long as theorder of the factors is not changed.Let (A,o) be an algebra structure. An element lA is a left identity for o if la=a for every aA. Give an example of a 3-element structure with exactly two left identities. Define a right identity for (A,o). Prove that if (A,o) has a right identity r and a left identity l, then r=l, and that r=l is an identity for o.Let G be a group. Prove that if a2=e for all aG, then G is abelian.Give an example of an algebraic structure of order 4 that has both right and left cancellation but that is not a group.Prove that an ordered field F is complete iff every nonempty subset of F that has a lower bound in F has an infimum in F.Prove that every irrational number is "missing" from . Begin with an irrational number x and find a subset A of such that A is bounded above in and sup(A) does not exist in , but when A is considered a subset of ,sup(A)=x.Find two upper bounds (if any exits) for each of the following sets. {x:x210} {13x:x} {x:x+1x5} {x:x2+2x30} {x:x230} {x:x0andx230} {2x:x} {x:x10logx}13E 14E 15E Let A and B be subsets of . Prove that if A is bounded above and BA, then B is bounded above. if A is bounded below and BA, then B is bounded below. if A and B are bounded above, then AB is bounded above. if A and B are bounded below, then AB is bounded below.17E 18E Give an example of a set A for which both A and Ac are unbounded above and below.

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