   Chapter 1.1, Problem 11E

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Textbook Problem
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# State the most general conditions on the subsets A     and  B     of     U under which the given equality holds. (a) A     ∩     B     =     A (b) A     ∪     B '     =     A (c) A     ∪     B     =     A (d) A     ∩     B '     =     A (e) A     ∩     B     =     U (f) A '     ∩     B '     =     ∅ (d) A     ∪     ∅     =     ∪ (h) A '     ∩     U     =     ∅

(a)

To determine

The most general conditions on the subsets A and B of U under which the equality AB=A holds.

Explanation

Given information:

A and B are subsets of U.

Formula used:

• 1) 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}

• 2) Subset:

A and B be sets. A is called a subset of B if and only if every element of A is an element of B

(b)

To determine

The most general conditions on the subsets A and B of U under which the equality AB'=A holds.

(c)

To determine

The most general conditions on the subsets A and B of U under which the equality AB=A holds.

(d)

To determine

The most general conditions on the subsets A and B of U under which the equality AB'=A holds.

(e)

To determine

The most general conditions on the subsets A and B of U under which the equality AB=U holds.

(f)

To determine

The most general conditions on the subsets A and B of U under which the equality A'B'=ϕ holds.

(g)

To determine

The most general condition on the subset A of U under which the equality

Aϕ=U holds.

(h)

To determine

The most general condition on the subset A of U under which the equality

A'U=ϕ holds.

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