Question

Using logical equivalences, show the following set identity:

A\(A∩B) =A\B .

Recall  that  to  show  this  set  identity,  you  have  to  derive  the  following  logical  equivalence,where x is an arbitrary element:

x∈A\(A∩B)⇐⇒x∈A\B .

Answer 1

1. {x∈A⋂B}⇒{x∈A AND x∈B}
2. {x∈A\B}⇒{x∈A AND x∉B}
3. {x∈A\(A⋂B)}⇒{x∈A AND x∉A⋂B} ({x∉A⋂B} contains only elements of A which are not in B)
4. {x∈A\(A⋂B)}⇒{x∈A AND x∉B} (retains elements of A which were not also in B)
5. {x∈A\(A⋂B)}⇒{x∈A\B} QED

EXAMPLE

A={ 1, 4, 6, 9, 10 }, B={2, 4, 6, 8 }, A\B={ 1, 9, 10 }, A⋂B={ 4, 6 }, A\(A⋂B)={ 1, 9, 10 }.