Question

If a+b=2
Prove
(a-b)^2 = (b-a)^2

Answer 1

Step-by-step explanation:

a+b=2

a=2-b

(2-b-b)^2=(b-2+b)^2

(2-2b)^2=(2b-2)^2

(-2(1+b))^2=(2(1+b))^2

4(1+b)^2=4(1+b)^2. proved

Answer 2

Answer:

Abelian Group problems and solutions

Problem 401

Let G be a group. Suppose that

(ab)2=a2b2

for any elements a,b in G. Prove that G is an abelian group.

Add to solve later

Proof.

To prove that G is an abelian group, we need

ab=ba

for any elements a,b in G.

By the given relation, we have

(ab)2=a2b2.

The left hand side is

(ab)2=(ab)(ab),

and thus the relation becomes

(ab)(ab)=a2b2.

Equivalently, we can express it as

abab=aabb.

Multiplying by a−1 on the left and b−1 on the right, we obtain

a−1(abab)b−1=a−1(aabb)b−1.

Since a−1a=e,bb−1=e, where e is the identity element of G, we have

ebae=eabe.

Since e is the identity element, it yields that

ba=ab

and this implies that G is an abelian group