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 ∪ ( C − B . )

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

if A∩C⊆B∩C and A∪C⊆B∪C then A=B.

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

If A∩C⊆B∩C and A∪C⊆B∪C then A=B.

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

Recall, the definition for union, and intersections,

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

1. The union of A and B is denoted by A∪B. And is defined by

A∪B={x∈U|x∈A, or, x∈B}

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}.

3. For a set A to be a subset of a set B is denoted by A⊆B, and is defined by

   A⊆B:{∀x ifx∈A,then, x∈B}

Let A,B and C be any sets.

Suppose, if x∈A when x∈C or x∉C.

Case 1.

For x∈C

Use the fact that A∩C⊆B∩C