For the pair of sets, explain if it is a subset of the other.

Question

A - B and A - (B - C)

Answer 1

First we define P-Q as P∩Q'
where ∩ is intersection, and Q' is the complement of Q'.
We will use de Morgan's law applied to sets as:
(P ∩ Q)' = P' ∪ Q'

Following this definition,
A-B = A∩B'

A-(B-C)
= A - (B∩C')
= A ∩ (B∩C')'
= A ∩ (B' ∪ C) (de Morgan's law)
= A∩B' ∪ A∩C (distributivity)

From which it is evident that A-B is a subset of A-(B-C).

Use a Venn diagram to visualize the above results.