Answer to Question #244604 in Discrete Mathematics for Alina

Question

Given two sets A and B, for each of the following statements, what can you conclude about the sets?

For example: consider the statement A−B=∅, this could be possible if -

Scenario - 1: if A=B then A−B=∅

Scenario - 2: Since A−B=A−(A∩B), if (A∩B) = A then A−B=∅

Scenario - 3: A=∅ in which case, no matter what B is A−B=∅.

Therefore we can conclude that if A−B=∅, then one of the above scenarios must be true. You do not need to draw exactly 3 conclusions. Try to answer with an exhaustive list of conclusions you can draw from each of the following statements:

(a) A∪B=A

(b) A∩B=A

(c) A−B=A

(d) A∩B=B∩A

(e)A−B=B−A

Answer 1

a) If A∪B=A, then

$B\subseteq A$

b) If A∩B=A, then

$A\subseteq B$

c) If A−B=A, then

$B=\empty$

d) If A∩B=B∩A, then

A and B are any sets.

e) If A−B=B−A, then

$A=B$

or

$A=\empty$

or

$B=\empty$