   Chapter 1.1, Problem 29E

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# In Exercises 14-35, prove each statement. A     ∪     ( B     −     A )     =     A     ∪     B

To determine

To prove: A(BA)=AB

Explanation

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) Complement:

For arbitrary subsets A and B of the universal set U, the complement of B in A is AB={xU|xAandxB}

3) Distributive law:

A(BC)=(AB)(AC)

A(BC)=(AB)(AC)

Proof:

Use definition of complement, the complement of A in B is given by,

BA={xU|xBandxA}

By definition of union,

A(BA)={xA}or{xU|xBandxA}

={xU|

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