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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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In Exercise 14-35, prove each statement.

A ( A ' B ) = A B

To determine

To prove: A(A'B)=AB

Explanation

Formula used:

1) Equality of sets:

If AB and BA then, A=B, for any sets A and B.

2) Subset:

A and B are sets, A is called the subset of B if and only if every element of A is an element of B.

3) Union:

If A and B are sets, the union of A and B is the set AB ( read as‘ A union B ’) given by

AB={x|xAorxB}.

4) Intersection:

If A and B are sets, the intersection of A and B is the set AB ( read as‘ A intersection B ’) given by, AB={x|xAandxB}

5) Complement:

A' is defined as A'=UA={xU|xA} the set A' ( read as complement of A ’)

Where U is the universal set.

6) Distributive law:

A(BC)=(AB)(AC)

A(BC)=(AB)(AC)

Proof:

Let xA(A'B)

xA and x(A'B) … (By definition of intersection of sets)

x

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