Let A = Z and let a, b ∈ A. Prove if the following binary operations...

Let A = Z and let a, b ∈ A. Prove if the following binary operations are (i) commutative, (2) if they are associative and (3) if they have an identity (if the operations has an identity, give the identity or show that the operation has no identity).

(a) (3 points) f(a, b) = a + b + 1

(b) (3 points) f(a, b) = a(b + 1)

Solutions

Expert Solution

Colby Messinger answered 5 months ago

Next > < Previous

Related Solutions

Let A = {a+b*sqrt14: a,b∈Z}. Prove that A ∩ Q = Z. Explain is set A...

Let A = {a+b*sqrt14: a,b∈Z}. Prove that A ∩ Q = Z. Explain is set A countable?

let n belongs to N and let a, b belong to Z. prove that a is...

let n belongs to N and let a, b belong to Z. prove that a is congruent to b, mod n, if and only if a and b have the same remainder when divided by n.

(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2....

(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2. Prove that if 2p−1 is prime, then p must also be prime. (Abstract Algebra)

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove...

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove that xa+b = xaxb b) prove that (xa)-1 = x-a c) establish part a) for arbitrary integers a and b in Z (positive, negative or zero)

Prove the following: (a) Let A be a ring and B be a field. Let f...

Prove the following: (a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal. (b) If A/J is a field, then J is a maximal ideal.

Suppose A = {(a, b)| a, b ∈ Z} = Z × Z. Let R be...

Suppose A = {(a, b)| a, b ∈ Z} = Z × Z. Let R be the relation define on A where (a, b)R(c, d) means that 2 a + d = b + 2 c. a. Prove that R is an equivalence relation. b. Find the equivalence classes [(−1, 1)] and [(−4, −2)].

a.) Prove the following: Lemma. Let a and b be integers. If both a and b...

a.) Prove the following: Lemma. Let a and b be integers. If both a and b have the form 4k+1 (where k is an integer), then ab also has the form 4k+1. b.)The lemma from part a generalizes two products of integers of the form 4k+1. State and prove the generalized lemma. c.) Prove that any natural number of the form 4k+3 has a prime factor of the form 4k+3.

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove...

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove (a). (a^m)(a^n)=a^(m+n) (b). (a^m)^n=a^(mn)

Bezout’s Theorem and the Fundamental Theorem of Arithmetic 1. Let a, b, c ∈ Z. Prove...

Bezout’s Theorem and the Fundamental Theorem of Arithmetic 1. Let a, b, c ∈ Z. Prove that c = ma + nb for some m, n ∈ Z if and only if gcd(a, b)|c. 2. Prove that if c|ab and gcd(a, c) = 1, then c|b. 3. Prove that for all a, b ∈ Z not both zero, gcd(a, b) = 1 if and only if a and b have no prime factors in common.

(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b)...

(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b) Prove that if m is composite, then Z/mZ is not a field. (4) Let m be an odd positive integer. Prove that every integer is congruent modulo m to exactly one element in the set of even integers {0, 2, 4, 6, , . . . , 2m− 2}

Subjects