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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
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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