For each of 5-21 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 and B, If A ⊆ B then A ∩ B e = ∅ .

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 and B, if A⊆B then A∩Bc=ϕ.

Given information:

Let  A,B    be any set

Concept used:

∪:Union of sets∩:Intersection of sets

⊆: subset of set

Calculation:

Consider the statement,

If A⊆B then A∩Bc=∅

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

Recall, the definition for Unions, and, Intersections.

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

1. For a set A to be a subset of a set B is denoted by A⊆B, and is defined by A⊆B;{∀x, if x∈A then x∈B}.
2. The complement of A is denoted by Ac and is defined by

Ac={x∈U|x∉A}.

The intersection of A and B, is denoted by A∩B, and is defined by A∩B={x∈U|x∈A and x∈B}

Prove that whether the statement is true or false as,

Let A,B be any sets and given that A⊆B