For each of 5-21 prove each statement that I true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U.
For all sets A,B, and C,
A ∩ ( B − C ) = ( A ∩ B ) − ( A ∩ C ) .

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U.

For all sets A,B and C.

A∩(B−C)=(A∩B)−(A∩C).

Given information:

Let  A,B  and C  be any set

Concept used:

∪:Union of sets∩:Intersection of sets

⊆: subset of set

Calculation:

Consider the statement,

A∩(B−C)=(A∩B)−(A∩C)

For all sets A,B and C.

Here all sets are subsets of a universal set U.

The objective is to verify whether the statement is true or not.

Recall, the definition for Differences, and, Intersections.

Suppose A and B are two subsets of a universal set U.

1. The difference of B minus A is denoted by B−A and is defined by, B−A={x∈U|x∈B, and x∉A}
2. The intersection of A and B is denoted by A∩B, and is defined by

A∩B={x∈U|x∈A, and x∈B}

Let A,B and C be any sets

Consider the left hand side

A∩(B−C)

Let x∈A∩(B−C)

⇒x∈A and x∈(B−C) intersection of two sets⇒x∈A and x∈B and x∉C Difference of two sets⇒x∈(A∩B) and x∉(A∩C) Intersection of two sets⇒x∈(A∩B)−(A∩C) Difference of two sets

Thus,

A∩(B−C)⊆(A∩B)−(A∩C)