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All Textbook Solutions for A Transition to Advanced Mathematics
20E21E22EFind 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}Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. Let A. If i=inf(A) and =0, then there is yA. such that yi+. "proof." Let y=i+2. Then iy, so yA. By construction of y,yi+. Claim. Let A. If A is bounded above, then Ac is bounded below. "proof." If A is bounded above, then sup(A) exists (because is complete). Because sup(A)=inf(Ac) (see the figure), inf(Ac) exists. Thus, Ac is bounded below. Claim. If AB, and if sup(A) and sup(B) both exist, then sup(A)sup(B). "proof." Assume that B and sup(A)sup(B). We choose =12(sup(A)sup(B)). Then 0 and sup(B)sup(A)sup(A). By part (ii) of Theorem 7.1.1, there is yA such that ysup(A). Then yB and ysup(B). This is impossible. Therefore, sup(A)sup(B). Claim. If f: and A is a bounded subset of , then the image setf(A) is bounded. "proof." Let m be an upper bound for A. Then am for all aA. Therefore, f(a)f(m) for all aA. Thus, f(m) is an upper bound for f(A).,/li> Claim. The set is not bounded above. "Proof." Suppose that is bounded above by the natural number m. Then m+1. is a natural number larger than m. This is a contradiction.3E4ELet 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 ABis bounded above. if A and B are bounded below, then AB is bounded below.Let x be an upper bound for A. Prove that if xy, then y is an upper bound for A. if xA, then x=sup(A).Let A. Prove that if A is bounded above, then Ac is not bounded above. if A is bounded below, then Ac is not bounded below.Give an example of a set A for which both A and Ac are unbounded above and below.Let A. Prove that if sup(A) exists, then it is unique. That is, if x and y are both least upper bounds for A, then x=y. if inf(A) exists, then it is unique.Formulate and prove a characterization of greatest lower bounds similar to that in Theorem 7.1.2 for least upper bounds.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.17EProve 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.Let A and B be compact subsets of . Use the definition of compact to prove that AB is compact. Apply the Heine-Borel Theorem to prove that AB is compact. Apply the Heine-Borel Theorem to prove that AB is compact.2E3E4EAssign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. Let a. The open ray (a,) is an open set. "Proof." Write the set (a,) as (a,)=n=0(a+n,a+n+1). Each set (a+n,a+n+1) is an open interval (each of length 1) and therefore an open set. By Theorem 7.2.2, (a,) is an open set. Claim. If A and B are compact, then AB is compact. "Proof." If A and B are compact, then for any cover {O:} for A, there exists a finite subcover O1,O2,...,On, and for any open cover {U:} for B, there exists a finite subcover U1,U2,...,Um. Thus, AO1O2On and BO1O2Om. Therefore, ABO1O2OnU1U2m, aunionof a finite number of open sets. Thus, AB is compact. Claim. If A is compact, BA, and B is closed, then B is compact. "Proof." Let {O:} be a cover for B. If {O:} is a cover for A, then there is a finite subcover of {O:} that covers A and hence covers B. If {O:} is not a cover for A, add one more open set O*=B to the collection to obtain a cover for A. This cover for A has a finite subcover of A that is a cover for B. In either case, B is covered by a finite number of open sets. Therefore, B is compact. Claim. If A is compact, BA, and B is closed, then B is compact. "Proof." B is closed by assumption. Because A is compact, A is bounded. Because BA, B is also bounded. Thus, B is closed and bounded. Therefore, B is compact. Claim. The set (5,) is compact. "Proof." The set l={(4,12),(10,)} is a cover for (5,). Then l is a finite subcover of l for (5,), so (5,), is compact.For real numbers x,1,2,...n, describe i=1nN(x,i). i=1nN(x,i).State the definition of continuity of the function fat the number a in terms of neighborhoods.Find the set of interior point for each of these subsets of . (1,1) (1,1] {13k:k} {13k:k}{0} {13k:k} n(n+0.1,n+0.2)Suppose that x is an interior point of a set A. Prove that if AB, then x is an interior point of B. if x is an interior point of B, then x is an interior point of AB. if x is an interior point of B, then x is an interior point of AB.Let AB. Prove that if sup(A) and sup(B) both exist, then sup(A)sup(B). if inf(A) and inf(B) both exist, then inf(A)inf(B).Let Abe a nonempty collection of closed subsets of . Prove that AAA is a closed set. Prove that if A is a finite collection, then AAA is a closed set. Show by example that part (b) is false if we do not assume that A is finite.12E13E14E15E16EProve Lemma 7.2.4.Which of the following subsets of are compact? [0,10][20,30] [,10] A, where A is finite set {1,2,3,4,9,12,18} {0}{1n:n} (3,5] [0,1]Give an example of a bounded subset of and a cover of that set that has no finite subcover. a closed subset of and a cover of that set that has no finite subcover. sets A, B, C, and D of real numbers such that ABCD, A is open, B is closed, C is neither open nor closed, and D is compact.Let A and F be sets of real numbers, and let F be finite. Prove that if x is an accumulation point of A, then x is an accumulation point of AF.In the proof of Theorem 7.3.1 that =, it is asserted that to prove that every interval (x,y) contains rationals, it suffices to consider the case when x and y are both positive. Explain.Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. For A, B,(AB)=AB. "Proof." Because AAB,A(AB) by Exercise 6(a). Likewise, B(AB). Therefore, AB(AB). To show that (AB),AAB, let x(AB). Then for all 0,N(x,) contains a point of AB distinct from x. Restating this, we say, for all 0, that N(x,) contains a point of A distinct from x or a point of B distinct from x. Thus, for all 0,N(x,) contains a point of A distinct from x, or for all 0,N(x,) contains a point of B distinct from x. But this means xA or xB. Therefore, xAB. Claim. For A,(Ac)=(A)c. “Proof.” x(A)c iff xA iff x is not an accumulation point for A iff x is an accumulation point for Ac iff x(Ac). Claim. For A, B,(AB)AB. “Proof.” ( AB)=( A B c )definitionofAB A( B c )Exercise7( b) A( B)Cbecause( B c ) ( B ) c =AB. Claim. If A is closed, then AA. "Proof." Suppose that A is closed. Then Ac is open. Let xAc. Then x is an interior point of Ac. Therefore, there exists 0, so N(x,)Ac. Hence, N(x,)A=. Thus, x is not an accumulation point for A. Because xAc implies xA, we conclude AA. Claim. If A is a set with an accumulation point, BA, and B is infinite, then B has an accumulation point. "Proof." First, A is infinite because BA and B is infinite. Because A has an accumulation point, by the Bolzano-Weierstrass Theorem A must be bounded. Because BA, this means B is bounded. Hence, by the Bolzano-Weierstrass Theorem again, B has an accumulation point.Prove that 7 is an accumulation point for [3,7). 5 is an accumulation point for (4,5)(5,6). 0 is an accumulation point for {1+( 1)nn:n}. e is an accumulation point for {(1+ 1 n)n:n}.Find an example of an infinite subset of that has no accumulation points. exactly one accumulation point. exactly two accumulation points.Find the derived set of each of the following sets. {n+12n:n} {2n:n} {6n:n} {72n:n} (0,1] (3,7){2,6,8} {1+( 1)nnn+1:n} (0,1) {1+n2(1+ ( 1 ) n)n:n} {sinx:x(2,2)} {k+1n:k,n} {x2y:y}Let S=(0,1]. Find S(Sc).8E(a)Prove that if AB, then AB. (b)Is the converse of part (a) true? Explain.Show by example that the intersection of infinitely many open sets may not be open.11E12ELet a, b. Prove that every closed interval [a,b] is a closed set.14E15E1EProve that if x is an interior point of the set A, then x is an accumulation point for A. Is the converse of part (a) true? Explain. Prove that if S is open, then every point of S is an accumulation point of S. Is the converse of part (c) true? Explain.Recall from Exercise 11 of Section 4.6 that the sequence yn is a subsequence of xn if and only if there is an increasing function f: such that yn=xf(n). Prove that if x is bounded, then every subsequence of x is bounded. if x is monotone, then every subsequence of x is monotone.A sequence x of real numbers is a Cauchy* sequence if for every 0, there exists an integer M such that if m,nM, then |xnxm|. That is, terms in the sequence are arbitrarily close together if the terms are chosen far enough along the sequence. Prove that if x is a Cauchy sequence, then x is bounded. Prove that if x is a convergent sequence, then x is a Cauchy sequence. (It can also be shown that every Cauchy sequence converges.)5EAssign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. Every bounded non-increasing sequence converges. "Proof." Let x be a bounded non-increasing sequence. Then yn=xn defines a bounded non-decreasing sequence. By the proof of Theorem 7.4.3, limnyn=L for some L. Thus, limnxn=L. Claim. The sequence x, where xn=lnnn, converges. * Augustin Louis Cauchy (1789—1857) was a creative mathematician and pioneer in the efforts to bring rigor to the infinitesimal calculus. He was the first to define complex numbers as a pair of real numbers. Cauchy's name is associated with concepts and results in many fields of mathematics, includinggeometry, analysis, and mathematical physics. "Proof." Because lnnn for all natural numbers n, lnnn1. Therefore, x is a bounded sequence. The derivative of lnnn is 1lnnn2 which is less than 0 for every natural number n greater than e. Therefore, except for the first two terms, x is a decreasing sequence. Because x is bounded and decreasing, x converges.7EGive an example of a bounded sequence that is not convergent. an increasing sequence that is not convergent. a convergent sequence that is not monotone. a divergent sequence x such that the sequence whose nth term is |xn|converges. an increasing sequence bounded above by 2 that does not converge to 2. an increasing sequence that converges to 2.9ELet A and B be subsets of . Prove that (AB)=AB. Prove that (AB)AB. Find a counterexample for (AB)=AB.For the sequence y defined in the proof of Theorem 7.5.3, explain why the sequence is eventually constant if ym=sup(A) for some m.2E3ELet I be a sequence of intervals. Then for each n,In is an interval on the real line. If In+1In for all n, the sequence I is called a sequence of nested intervals. Give an example of a sequence I of nested open intervals such that nIn=. Give an example of a sequence I of nested open intervals such that nIn=(1,2]. (The Nested Interval Theorem) Use the results of this chapter to prove that if I isa sequence of nested closed intervals, then nIn is nonempty. Give an example of a sequence I of nested closed intervals such that nIn=[1,2]. Give an example of a sequence I of nested closed intervals such that nIn={1}.5E6EFind all divisors of zero in 14. 15. 10. 101.8ESuppose m and m2. Prove that 1 and m1 are distinct units in (m,).10E11EDetermine whether each sequence is monotone. For each sequence that is monotone, prove your answer. xn=n+2n yn=2n xn(n2)(n5)2 yn=10n! xn=2n(n+2n) xn=2n5n+3 yn=n!nn xn=n!n+1 xn=n+113EComplete the proof that xn=(1+1n)n is increasing by showing that xnxn+1 for all n.15E16E17E1ERepeat Exercise 2 with the operation * given by the table on the right.3ELet 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.10EComplete the proof of Theorem 6.1.4. First, show that 1 is an element of (Um,).12E13EGive an example of an algebraic structure of order 4 that has both right and left cancellation but that is not a group.Let G be a group. Prove that G is abelian if and only if a2b2=(ab)2 for all a,bG. G is abelian if and only if anbn=(ab)n for all n and a,b,G.3E(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.Let m be a prime natural number and a(Um,). Prove that a=a1 if and only if a=1 or a=m1.7E8E9EThe 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?2E3E4EGive an example of an algebraic structure of order 4 that has both right and left cancellation but that is not a group.6EShow 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.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(a)Prove that (m,+) is associative and commutative and has an identity and that every element has an additive inverse (Theorem 6.1.2(a)). (b)Prove that (m,) is associative and commutative and has an identity when m1 (Theorem 6.1.2(b)).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.15E16E17EConsider the set A={a,b,c,d} with operation ogiven by the Cayley tableat the right. Name the identity element of this system. Is the operation oassociative on A? Is the operation ocommutative on A? For each element of A that has an inverse,name the inverse. Is B1={a,b,c} closed under o? Is B2={a,c} closed under o? Name all subsets of A that are closed under o. True or False? For all x,yA,xx=yy.Repeat Exercise 2 with the operation * given by the table on the right.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.21E22EShow 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.Given that G={e,u,v,w} is a group of order 4 with identity e,u2=v, and v2=e, construct the operation table for G.3EGive 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.Construct the operation table for S2. Is S2 abelian?6ELet 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.9E10E11EAssign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A. Claim. If G is a group with identity e, then G is abelian. "Proof." Let a and b be elements of G. Then ab=aeb=a(ab)(ab)1b=a(ab)(b 1a 1)b=(aa)(bb 1)a1b=(aa)a1b=(aa)( b 1a)1=a(a ( b 1 a ) 1)=a( ( b 1 a ) 1a)=a(( a 1 b)a)=(aa 1)(ba)=e(ba)=ba. Therefore, ab=ba and G is abelian. Claim. If G is a group with elements x, y, and z and if xz=xy, then x=y. "Proof." If z=e, then xz=yz implies that xe=ye, so x=y. If ze, then the inverse of z exists, and xz=yz implies xzz=yzz and x=y. Hence, in all cases, if xz=yz, then x=y. Claim. The set + of positive rationals with the operation of multiplication is a group. "Proof." The product of two positive rationals is a positive rational, so + is closed under multiplication. Because 1r=r=r1 for every r,1 is the identity. The inverse of the positive rational g is the positive rational ba. The rationals are associative under multiplication because the reals are associative under multiplication. Claim. If m is prime, then (m{0},) has no divisors of zero. "proof." Suppose that a is a divisor of zero in (m{0},). Then a0, and there exists b0 in m such that ab=0. Then ab=m (mod m), so ab=m. This contradicts the assumption that m is prime. Claim. If m is prime, then (m{0},) has no divisors of zero. "proof." Suppose that a is a divisor of zero in (m{0},). Then a0, and there exists b0 in m such that ab=0. Then ab=0 (mod m), so m divides ab. Because m is prime, m divides a, or m divides b. But because a and b are elements of m, both are less than m. This is impossible. Claim. For every natural number m, (m{0},) is a group. "Proof." We know that (,) is associative with identity element 1. Therefore, (m{0},) is associative with identity element 1. It remains to show every element has an inverse. For xm{0},x0. Therefore, 1/xm{0} and x1/x=x(1/x)=1. Therefore, every element of m{0} has an inverse. Claim. If (m{0},) is a group, then m is prime. "Proof." Assume that (m{0},) is a group. Suppose that m is not prime. Let m=rs, where r and s are integers greater than 1 and less than m. Then rs=m=0 (mod m). Because r has an inverse t in m{0},tr=1. Then s=1s=(tr)s=t(rs)=t0=0. That is, s=0 (mod m). This is impossible because 1sm.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)).Prove that if H and K are subgroups of a group G, then HK is a sub- group of G. Prove that if {H:} is a family of subgroups of a group G, then H is a subgroup of G. 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.7E8E9EList all generators of each cyclic group in Exercise 9.11ELet 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 .14E15ELet 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.Is S3 isomorphic to (6,+)? Explain.2EUse the method of proof of Cayley's Theorem to find a group of permutations isomorphic to (3,+). (5,+). (,+).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)).6EDefine 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.9EFind the order of each element of the group S3. (7,+). (8,+). (U11,).11ELet (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.13E14E15E16E17E1E2EShow that any two groups of order 2 are isomorphic. Show that any two groups of order 3 are isomorphic. Prove that there exist two groups of order 4 that are not isomorphic.Show that the function h: defined by h(x)=3x is not a ring homomorphism. Show that the function h:66 defined by h(x)=2x is not a ring homomorphism. 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.6E7ELet (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.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.1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16EShow 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 complex number multiplication., 1,1,i,i,, where is complex number multiplication. PX,, where X is a nonempty set and is the symmetric difference operation AB=ABBA. t,+, where t is a natural number. The set of 22 real matrices with determinant 1, where the operation is matrix multiplication.2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E1E2E3E4E5E6E7E8E9E10E11E12E13EProve that for every natural number m greater than 1, the group m,+ is cyclic with generators 1 and m1.Prove that every subgroup of a cyclic group is cyclic.16E17E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16EIs S3 isomorphic to 6,+? Explain.Prove that the relation of isomorphism is an equivalence relation. That is, prove that (a) if G, is a group, then G, is isomorphic to G,. (b) if G, is isomorphic to H,* then H,* is isomorphic to G,. (c) if G, is isomorphic to H,* and H,* is isomorphic to K,, then G, is isomorphic to K,.19E20E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15EAn 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 0n1n. Prove that the two versions are equivalent.17E18E19E20E1E2E3E4E5E6E7E8E9E10E11E12E