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All Textbook Solutions for Elements Of Modern Algebra
True or False Label each of the following statements as either true or false. Two sets are equal if and only if they contain exactly the same elements.True or False
Label each of the following statements as either true or false.
2. If is a subset of and is a subset of , then and are equal.
True or False
Label each of the following statements as either true or false.
3. The empty set is a subset of every set except itself.
True or False Label each of the following statements as either true or false. AA= for all sets A.5TFETrue or False Label each of the following statements as either true or false. AA for all sets A.True or False
Label each of the following statements as either true or false.
7.
True or False
Label each of the following statements as either true or false.
8.
True or False Label each of the following statements as either true or false. AB=CB implies A=C, for all sets A,B, and C.True or False Label each of the following statements as either true or false. AB=AC implies B=C, for all sets A,B, and C.1E2. Decide whether or not each statement is true for and .
a. b.
c. d.
e. f.
Decide whether or not each statement is true. (a) a{a,{a}} (b) {a}{a,{a}} (c) {a}{a,{a}} (d) {{a}}{a,{a}} (e) A (f) 0 (g) {} (h) {} (i) {} (j) {}= (k) (l)4. Decide whether or not each of the following is true for all sets .
a. b.
c. d.
e. f.
g. h.
i. j.
k. l.
5E6. Determine whether each of the following is either , , , or , where is an arbitrary subset of the universal set .
a. b.
c. d.
e. f.
g. h.
i. j.
k. l.
m. n.
7E8. Describe two partitions of each of the following sets.
a. b.
c. d.
9E10E11E12. Let Z denote the set of all integers, and let
Prove that .
13. Let Z denote the set of all integers, and let
Prove that .
14E15EIn Exercises , prove each statement.
16. If and , then .
In Exercises , prove each statement.
17. if and only if .
In Exercises , prove each statement.
18.
19EIn Exercises 1435, prove each statement. (AB)=AB21E22EIn Exercises 14-35, prove each statement.
23.
24EIn Exercise 14-35, prove each statement. If AB, then ACBC.In Exercise 14-35, prove each statement.
26. If then .
In Exercise 14-35, prove each statement.
27.
28EIn Exercises 14-35, prove each statement.
29.
In Exercises 14-35, prove each statement. (AB)C=(AC)(BC)In Exercises 1435, prove each statement. (AB)(AB)=AIn Exercises 1435, prove each statement. U(AB)=(UA)(UB)In Exercises , prove each statement.
33.
In Exercises , prove each statement.
34. if and only if
In Exercises 1435, prove each statement. AB if and only if AB=A.Prove or disprove that AB=AC implies B=C.Prove or disprove that AB=AC implies B=C.38. Prove or disprove that .
39E40. Prove or disprove that .
Express (AB)(AB) in terms of unions and intersections that involve A,A,B,andB42. Let the operation of addition be defined on subsets by. Use a Venn diagram with labelled regions to illustrate each of the following statements.
a.
b.
c. .
43. Let the operation of addition be as defined in Exercise 42. Prove each of the following statements.
a. b.
42. Let the operation of addition be defined on subsets and of by
Label each of the following statements as either true or false.
1. , for every nonempty set A.
Label each of the following statements as either true or false.
2. for all nonempty sets A and B.
Label each of the following statements as either true or false.
3. Let where A and B are nonempty. Then for every subset S of A.
Label each of the following statements as either true or false. Let f:AB where A and B are nonempty. Then f1(f(T))=T for every subset T of B.Label each of the following statements as either true or false. Let f:AB. Then f(A)=B for all nonempty sets A and B.True or False
Label each of the following statements as either true or false.
6. Every bijection is both one-to-one and onto.
Label each of the following statements as either true or false. A mapping is onto if and only if its codomain and range are equal.Label each of the following statements as either true or false.
8. Let and . Then for every in .
Label each of the following statements as either true or false.
9. Composition of mappings is an associative operation.
1EFor each of the following mapping, state the domain, the codomain, and the range, where f:EZ. f(x)=x2,xE f(x)=x,xE f(x)=| x |,xE f(x)=x+1,xE3. For each of the following mappings, write out and for the given and, where.
For each of the following mappings f:ZZ, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. a. f(x)=2x b. f(x)=3x c. f(x)=x+3 d. f(x)=x3 e. f(x)=|x| f. f(x)=x|x| g. f(x)={xifxiseven2x1ifxisodd h. f(x)={xifxisevenx1ifxisodd i. f(x)={xifxisevenx12ifxisodd j. f(x)={x1ifxiseven2xifxisodd5. For each of the following mappings, determine whether the mapping is onto and whether it is one-to-one. Justify all negative answers. (Compare these results with the corresponding parts of Exercise 4.)
a. b.
c. d.
e. f.
6. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers.
a. b.
7. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers.
a. b.
c. d.
8. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers.
a. b.
For the given subsets A and B of Z, let f(x)=2x and determine whether f:AB is onto and whether it is one-to-one. Justify all negative answers. a. A=Z+,B=Z b. A=Z+,B=Z+EFor each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not ontoFor the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisodd12. Let and . For the given , decide whether is onto and whether it is one-to-one. Prove that your decisions are correct.
a. b.
c. d.
13. For the given decide whether is onto and whether it is one-to-one. Prove that your decisions are correct.
a.
b.
c.
d.
e.
f.
g.
h.
14. Let be given by
a. Prove or disprove that is onto.
b. Prove or disprove that is one-to-one.
c. Prove or disprove that .
d. Prove or disprove that .
15. a. Show that the mapping given in Example 2 is neither onto nor one-to-one.
b. For this mapping , show that if , then .
c. For this same and , show that .
16. Let be given by
a. For , find and .
b. For find and.
17. Let be given by
a. For find and.
b. For find and.
18. Let and be defined as follows. In each case, compute for arbitrary .
a.
b.
c.
d.
e.
19E20EIn Exercises 20-22, Suppose and are positive integers, is a set with elements, and is a set with exactly elements.
21. If, how many one-to-one correspondences are there from to?
22ELet a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.24. Let, where and are nonempty.
Prove that for all subsets and of.
Prove that for all subsets and of.
Give an example where there are subsets and of such that.
Prove that for all subsets and of.
Give an example where there are subsets and of such that
.
25. Let, where and are non empty, and let and be subsets of .
Prove that.
Prove that.
Prove that.
Prove that if.
26. Let and. Prove that for any subset of T of .
27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.).
15.
b. For the mapping , show that if , then .
28. Let where and are nonempty. Prove that has the property that for every subset of if and only if is onto. (Compare with Exercise 15c.)
Exercise 15c.
c. For this same and show that.
Label each of the following statements as either true or false.
1. Mapping composition is a commutative operation.
Label each of the following statements as either true or false. The composition of two bijections is also a bijection.Label each of the following statements as either true or false.
3. Let , , and be mappings from into such that . Then .
Label each of the following statements as either true or false.
4. Let , , and be mappings from into such that . Then .
True or False
Label each of the following statements as either true or false.
Let and such that is onto. Then both and are onto.
Label each of the following statements as either true or false. Let g:AB and f:BC such that fg is one-to-one. Then both f and g are one-to-one.For each of the following pairs and decide whether is onto or one-to-one and justify all negative answers.
For each pair given in Exercise 1, decide whether is onto or one-to-one, and justify all negative answers.
Exercise 1:
Let . Find mappings and such that.
Give an example of mappings and such that one of or is not onto but is onto.
Give an example of mapping and different from those in Example 3, such that one of or is not one-to-one but is one-to-one.
Example 3.
Let . Find mappings and such that.
6. a. Give an example of mappings and , different from those in Example , where is one-to-one, is onto, and is not one-to-one.
b. Give an example of mappings and , different from Example , where is one-to-one, is onto, and is not onto.
7. a. Give an example of mappings and , where is onto, is one-to-one, and is not one-to-one.
b. Give an example of mappings and , different from example , where is onto, is one-to-one, and is not onto.
Suppose f,g and h are all mappings of a set A into itself. a. Prove that if g is onto and fg=hg, then f=h. b. Prove that if f is one-to-one and fg=fh, then g=h.Find mappings f,g and h of a set A into itself such that fg=hg and fh. Find mappings f,g and h of a set A into itself such that fg=fh and gh.Let g:AB and f:BC. Prove that f is onto if fg is onto.11. Let and . Prove that is one-to-one if is one-to-one.
Let f:AB and g:BA. Prove that f is one-to-one and onto if fg is one to-one and gf onto.True or False
Label each of the following statements as either true or false.
1. If a binary operation on a nonempty set is commutative, then an identity element will exist in .
True or False
Label each of the following statements as either true or false.
2. If * is a binary operation on a nonempty set , then is closed with respect to *.
Label each of the following statements as either true or false. Let ={a,b,c}. The power set P(A) is closed with respect to the binary operation of forming intersections.True or False
Label each of the following statements as either true or false.
4. Let . The empty set is the identity element in with respect to the binary operation .
True or False Label each of the following statements as either true or false. Let A={ a,b,c }. The power set (A) is closed with respect to the binary operation of forming unions.True or False
Label each of the following statements as either true or false.
6. Let . The empty set is the identity element in power set with respect to the binary operation .
True or False
Label each of the following statements as either true or false.
7. Any binary operation defined on a set containing a single element is commutative and associative.
True or False
Label each of the following statements as either true or false.
8. An identity and inverses exist in a set containing a single element upon which a binary operation is defined.
True or False Label each of the following statements as either true or false. The set of all bijections from A to A is closed with respect to the binary operation of composition defined on the set of all mappings from A to A.1EIn each part following, a rule that determines a binary operation on the set of all integers is given. Determine in each case whether the operation is commutative or associative and whether there is an identity element. Also find the inverse of each invertible element.
b.
d.
f.
h.
j.
l.
for n. for
3E4E5E6E7. Prove or disprove that the set of nonzero integers is closed with respect to division defined on the set of nonzero real numbers.
8. Prove or disprove that the set of all odd integers is closed with respect to addition defined on , the set of integers.
9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to
a. addition defined on .
b. multiplication defined on .
10. Prove or disprove that the set of all nonzero integers is closed with respect to
a. addition defined on .
b. multiplication defined on .
11E12EAssume that is an associative binary operation on the non empty set A. Prove that a[ b(cd) ]=[ a(bc) ]d for all a,b,c, and d in A.Assume that is a binary operation on a non empty set A, and suppose that is both commutative and associative. Use the definitions of the commutative and associative properties to show that [ (ab)c ]d=(dc)(ab) for all a,b,c and d in A.15. Let be a binary operation on the non empty set . Prove that if contains an identity element with respect to , the identity element is unique.
Assume that is an associative binary operation on A with an identity element. Prove that the inverse of an element is unique when it exists.True or False Label each of the following statements as either true or false. Every permutation has an inverse.True or False Label each of the following statements as either true or false. Let A and f:AA. Then f is one to one if and only if f has a right inverse.3TFEFor each of the following mappings exhibit a right inverse of with respect to mapping composition whenever one exists.
a. b.
c. d.
e. f.
g. h.
i. j.
k. l.
m. n.
2. For each of the mappings given in Exercise 1, determine whether has a left inverse.
Exhibit a left inverse whenever one exists.
For each of the following mappings exhibit a right inverse of with respect to mapping composition whenever one exists.
a. b.
c. d.
e. f.
g. h.
i. j.
k. l.
m. n.
3E4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.6. Prove that if is a permutation on , then is a permutation on .
Prove that if f is a permutation on A, then (f1)1=f.8. a. Prove that the set of all onto mappings from to is closed under composition of mappings.
b. Prove that the set of all one-to-one mappings from to is closed under composition of mappings.
Let f and g be permutations on A. Prove that (fg)1=g1f1.10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.
Label each of the following statements as either true or false. Matrix addition is a binary operation from Mmn(R)Mmn(R) to Mmn(R).Label each of the following statements as either true or false. Matrix multiplication is a binary operation from Mmn(R)Mmn(R) to Mmn(R).3TFE4TFE5TFE6TFE7TFE8TFE9TFE10TFE11TFELabel each of the following statements as either true or false. Let A be a square matrix of order n over R such that A23A+In=On. Then A1=3InA.Write out the matrix that matches the given description.
2E3. Perform the following multiplications, if possible.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
Let A=[aij]23 where aij=i+j, and let B=[bij]34 where bij=2ij. If AB=[cij]24, write a formula for cij in terms of i and j.5E6ELet ij denote the Kronecker delta: ij=1 if i=j, and ij=0 if ij. Find the value of the following expressions. a. i=1n(j=1nij) b. i=1n(j=1n(1ij)) c. i=1n(j=1n(ij)ij) d. j=1nijjk8E9EFind two nonzero matrices A and B such that AB=BA.11. Find two nonzero matrices and such that.
12. Positive integral powers of a square matrix are defined by and for every positive integer. Evaluate and and compare the results for and.
13E14E15. Assume that are in and with and invertible. Solve for.
16E17EProve part b of Theorem 1.35.
Theorem 1.35 Special Properties of
Let be an arbitrary matrix over. With as defined in the preceding paragraph,
19E20ESuppose that A is an invertible matrix over and O is a zero matrix. Prove that if AX=O, then X=O.Let be the set of all elements of that have one row that consists of zeros and one row of the form with .
Show that is closed under multiplication in .
Show that for each in there is an element in such that .
Show that does not have an identity element with respect to multiplication.
Prove that the set S={[abba]|a,b} is closed with respect to matrix addition and multiplication in M2().24ELet A and B be square matrices of order n over Prove or disprove that the product AB is a diagonal matrix of order n over if B is a diagonal matrix.26EA square matrix A=[aij]n with aij=0 for all ij is called upper triangular. Prove or disprove each of the following statements. The set of all upper triangular matrices is closed with respect to matrix addition in Mn(). The set of all upper triangular matrices is closed with respect to matrix multiplication in Mn(). If A and B are square and the product AB is upper triangular, then at least one of A or B is upper triangular.28E29E30E31E32ELabel each of the following statements as either true or false. Every mapping on a nonempty set A is a relation.True or False
Label each of the following statements as either true or false.
2. Every relation on a nonempty set is as mapping.
True or False
Label each of the following statements as either true or false.
If is an equivalence relation on a nonempty set, then the distinct equivalence classes of form a partition of.
Label each of the following statements as either true or false. If R is an equivalence relation on a nonempty set A, then any two equivalence classes of R contain the same number of element.True or False
Label each of the following statements as either true or false.
Let be an equivalence relation on a nonempty setand let and be in. If, then.
Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.For determine which of the following relations onare mappings from
to, and justify your answer.
b.
d.
f.
2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric or transitive. Justify your answers.
a. if and only if .
b. if and only if .
c. if and only if for some in .
d. if and only if .
e. if and only if .
f. if and only if .
g. if and only if .
h. if and only if .
i. if and only if .
j. if and only if .
k. if and only if .
a. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the equivalence class [ 3 ]. b. Let R be the equivalence relation congruence modulo 4 that is defined on Z in Example 4. For this R, list five members of equivalence class [ 7 ].4. Let be the relation “congruence modulo 5” defined on as follows: is congruent to modulo if and only if is a multiple of , and we write .
a. Prove that “congruence modulo ” is an equivalence relation.
b. List five members of each of the equivalence classes and .
5. Let be the relation “congruence modulo ” defined on as follows: is congruent to modulo if and only if is a multiple of , we write .
a. Prove that “congruence modulo ” is an equivalence relation.
b. List five members of each of the equivalence classes and .
In Exercises 610, a relation R is defined on the set Z of all integers, In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and least four members of each. xRy if and only if x2+y2 is a multiple of 2.In Exercises 610, a relation R is defined on the set Z of all integers, In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and least four members of each. xRy if and only if x2y2 is a multiple of 5.In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each. xRy if and only if x+3y is a multiple of 4.In Exercises 610, a relation R is defined on the set Z of all integers, In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each. xRy if and only if 3x10y is a multiple of 7.In Exercises , a relation is defined on the set of all integers. In each case, prove that is an equivalence relation. Find the distinct equivalence classes of and list at least four members of each.
10. if and only if .
Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and justify your decisions.
if and only ifand are parallel.
if and only ifand are perpendicular.
13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers.
a. if and only if is subset of .
b. if and only if is a proper subset of .
c. if and only if and have the same number of elements.
In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.Let A=R0, the set of all nonzero real numbers, and consider the following relations on AA. Decide in each case whether R is an equivalence relation, and justify your answers. (a,b)R(c,d) if and only if ad=bc. (a,b)R(c,d) if and only if ab=cd. (a,b)R(c,d) if and only if a2+b2=c2+d2. (a,b)R(c,d) if and only if ab=cd.16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.
In each of the following parts, a relation R is defined on the power set (A) of the nonempty set A. Determine in each case whether R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if xy. b. xRy if and only if xy.Let (A) be the power set of the nonempty set A, and let C denote a fixed subset of A. Define R on (A) by xRy if and only if xC=yC. Prove that R is an equivalence relation on (A).For each of the following relations R defined on the set A of all triangles in a plane, determine whether R is reflexive, symmetric, or transitive. Justify your answers. a. aRb if and only if a is similar to b. b. aRb if and only if a is congruent to b.Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive?
2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers.
a. if and only if
b. if and only if
c. if and only if for some in .
d. if and only if
e. if and only if
f. if and only if
g. if and only if
h. if and only if
i. if and only if
j. if and only if.
k. if and only if.
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.23EFor any relation on the nonempty set, the inverse of is the relation defined by if and only if . Prove the following statements.
is symmetric if and only if .
is antisymmetric if and only if is a subset of .
is asymmetric if and only if .
25E26EProve Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
28E29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there is a subset such that . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets .
Suppose thatis an onto mapping from to. Prove that if ℒ, is a partition of, then ℒ, is a partition of.
True or False Label each of the following statements as either true or false. The set Z of integers is closed with respect to subtraction.True or False Label each of the following statements as either true or false. The set ZZ+ is closed with respect to subtraction.True or False
Label each of the following statements as either true or false.
3. The set is closed with respect to multiplication.
True or False Label each of the following statements as either true or false. If xy=xz for all x,y, and z in Z, then y=z.True or False
Label each of the following statements as either true or false.
5. Let be a set of integers closed under subtraction. If and are elements of , then is in for any in .
6TFE7TFE8TFE9TFE10TFEProve that the equalities in Exercises 111 hold for all x,y,zandw in Z. Assume only the basic postulates for Z and those properties proved in this section. Subtraction is defined by xy=x+(y). x0=02E3E4EProve that the equalities in Exercises hold for all in . Assume only the basic postulates for and those properties proved in this section. Subtraction is defined by .
6E7EProve that the equalities in Exercises hold for all in . Assume only the basic postulates for and those properties proved in this section. Subtraction is defined by .
Prove that the equalities in Exercises hold for all in . Assume only the basic postulates for and those properties proved in this section. Subtraction is defined by .
10E11ELet A be a set of integers closed under subtraction. a. Prove that if A is nonempty, then 0 is in A. b. Prove that if x is in A then x is in A.13EIn Exercises , prove the statements concerning the relation on the set of all integers.
14. If and , then .
15E16EIn Exercises , prove the statements concerning the relation on the set of all integers.
17. If and , then .
In Exercises , prove the statements concerning the relation on the set of all integers.
18. If and , then .
In Exercises 13-24, prove the statements concerning the relation on the set of all integers.
19. If and, then.
In Exercises 1324, prove the statements concerning the relation on the set Z of all integers. If 0xy, then x2y2.21E22E23E24E25. Prove that if and are integers and, then either or.
(Hint: If, then either or, and similarly for. Consider for the various causes.)
Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.Let x and y be in Z, not both zero, then x2+y2Z+.28E29E30E31. Prove that if is positive and is negative, then is negative.
32. Prove that if is positive and is positive, then is positive.
33. Prove that if is positive and is negative, then is negative.
34E35EProve that the statements in Exercises are true for every positive integer .
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Prove that the statements in Exercises are true for every positive integer .
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3EProve that the statements in Exercises are true for every positive integer .
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5E6E7E8E9E10E11E12E13E14E15EProve that the statements in Exercises 116 are true for every positive integer n. a+ar+ar2++arn1=a1rn1rifr117. Use mathematical induction to prove that the stated property of the sigma notation is true for all positive integers . (This sigma notation is defined in Section .)
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b.
Let be integers, and let be positive integers. Use induction to prove the statements in Exercises . ( The definitions of and are given before Theorem in Section .)
18.
Let xandy be integers, and let mandn be positive integers. Use induction to prove the statements in Exercises 1823. ( The definitions of xn and nx are given before Theorem 2.5 in Section 2.1.) xmxn=xm+nLet xandy be integers, and let mandn be positive integers. Use induction to prove the statements in Exercises 1823. ( The definitions of xn and nx are given before Theorem 2.5 in Section 2.1.) (xm)n=xmnLet x and y be integers, and let m and n be positive integers. Use mathematical induction to prove the statements in Exercises 1823. ( The definitions of xn and nx are given before Theorem 2.5 in Section 2.1 ) n(x+y)=nx+nyLet x and y be integers, and let m and n be positive integers. Use mathematical induction to prove the statements in Exercises 1823. ( The definitions of xn and nx are given before Theorem 2.5 in Section 2.1 ) (m+n)x=mx+nxLet and be integers, and let and be positive integers. Use mathematical induction to prove the statements in Exercises. The definitions of and are given before Theorem in Section
24E25E26EUse the equation (nr1)+(nr)=(n+1r) for 1rn. And mathematical induction on n to prove 2n=(n0)+(n1)+(n2)+...+(nn) For all positive integers n.Use the equation. (nr1)+(nr)=(n+1r) for 1rn. andmathematical induction on n to prove the binomial theorem as it is stated in Exercise 25. 25. Let a and b be a real number, and let n be integers with 0rn. The binomial theorem states that (a+b)n=(n0)an+(n1)an1b+(n2)an2b2+...+(nr)anrbr+.......+(nn2)a2bn2+(nn1)abn1+(nn)bn =r=0n(nr)anrbr Where the binomial coefficients (nr) are defined by (nr)=n!(nr)!r!, With r!=r(r1).........(2)(1) for r1 and 0!=1. Prove that the binomial coefficients satisfy the equation (nr1)+(nr)=(n+1r) for 1rn This equation generates all the "interior "entries (printed in bold) of Pascal's triangle. 1111211331146411510105129E