Question

Let A and B be two sets. For each of the following equalities, say whether the quality is always true, never true, or sometimes true and sometimes false. In the justification, if the equality is always true or never true, briefly explain why. If the inequality is sometimes true and sometimes false, give specific examples where it is true and where it is false.

(a) A (A\B) - B\(BIA).
(b) A (BA) - B\(\B).

Answer 1

hello your question is incomplete below is the missing parts

(a) A\ (A\B) = B\(B\A)

(b) A\ (BA) = B\(A\B)

answer : A\ (A\B) = B\(B\A) = always true

A\ (BA) = B\(A\B) = sometimes true and sometimes false

Explanation:

(a) A\ (A\B) = B\(B\A). = ALWAYS TRUE

using de Morgan's law to prove this

A\ (A\B) = A\ ( A ∩ B^c )

= A ∩ ( A^C ∪ B )

= ( A ∩ A^C ) ∪ ( A ∩ B )

= Ф ∪ ( A ∩ B )

= ( A ∩ B )

ALSO : B\(B\A) = attached below is the remaining parts of the solution

B) A\ (BA) = B\(A\B) = Sometimes true and sometimes false

attached below is the prove using De Morgan's law