Simplify the given expression. Cite a property from Theorem 6.2.2 for each step. (A − (A ∩ B)) ∩ (B − (A ∩ B)) Let A and B be any sets. Then (A − (A ∩ B)) ∩ (B − (A ∩ B)) = = (A ∩ (A ∩ B)c) ∩ (B ∩ (A ∩ B)c) = A ∩ ((A ∩ B)c ∩ (B ∩ (A ∩ B)c)) = A ∩ (((A ∩ B)c ∩ B) ∩ (A ∩ B)c) = A ∩ ((B ∩ (A ∩ B)c) ∩ (A ∩ B)c) = A ∩ (B ∩ ((A ∩ B)c ∩ (A ∩ B)c)) = A ∩ (B ∩ (A ∩ B)c) = (A ∩ B) ∩ (A ∩ B)c = ∅

Step-by-step explanation:

Consider the sets A and B

(A − (A ∩ B)) ∩ (B − (A ∩ B))

= (A ∩ (A ∩ B)c) ∩ (B ∩ (A ∩ B)c) by the set difference law

= (A ∩ (Ac ∩ B)c) ∩ (B ∩ (Ac ∩ B)c) by De Morgan's law

= {(A ∩ Ac) ∪ (A ∩ Bc)} ∩ {(B ∩ Ac) ∪ (B ∩ Bc)} by the distributive law

= {∅ ∪ (A ∩ Bc)} ∩ {(B ∩ Ac) ∪ ∅} by complementation

= {A ∩ Bc} ∩ {B ∩ Ac} by identity law

= (A ∩ Ac) ∩ (B ∩ Ac) by the associative law

= ∅ ∩ ∅ by complementation

= ∅ by the universal bound law

Therefore,  (A − (A ∩ B)) ∩ (B − (A ∩ B)) = ∅

a function could be identified in 3 ways either on a chart,graph, or ordered pairs. first a function is that each point has its own. the rule is that there can be many x's to one y but there can not be 2 x's at different y's. so on a graph when you put a straight line through the y's there should be one dot only to be a function.

step-by-step explanation:

can you rephrase the question ? it is unclear : -)

step-by-step explanation: