   Chapter 1.1, Problem 41E

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# Express ( A     ∪     B )     −     ( A     ∩     B ) in terms of unions and intersections that involve A ,     A ' ,     B ,     and   B '

To determine

(AB)(AB) in terms of union and intersections that involves

A,A',B,B'

Explanation

Given information:

(AB)(AB)

Formula used:

1) 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}.

2) 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}.

3) Complement:

For arbitrary subsets A and B of the universal set U, the complement of B in A is

AB={xU|xAandxB}

A'=UA={xU|xA}

4) Distributive properties:

A(BC)=(AB)(AC)

A(BC)=(AB)(AC)

5) De Morgan’s laws:

(AB)'=A'B' and (AB)'=A'B'

Proof:

From the definition of union and intersection,

AB={x|xAorxB} and AB={x|xAandxB}

Now,

(AB)(AB)

={xU|xABandxAB}

By using definition of complement,

={xU|xABandx(A </

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