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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
121 views

Let Z denote the set of all integers, and let

A = { x | x = 3 p 2  for some  p Z }

B = { x | x = 3 q + 1  for some  q Z }

Prove that A = B .

To determine

To prove: The statement A=B where the two sets A and B are given by,

A={x|x=3p2 for some pZ} and B={x|x=3q+1 for some qZ}.

Explanation

Formula Used:

There are two sets A and B, if every element of A is an element of B then A is a subset of B. Symbolically, it is denoted as AB.

If x is an element of A then it is denoted as xA.

Proof:

The set A is given by,

A={x|x=3p2 for some pZ}

Rewrite x as,

x=3p2x=3(p1)+1

Substitute q for p1 in the above equation.

x=3q+1

So, xAxB

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