Use an element argument to prove each statement. Assume that all sets are subsets of a universal set U.

For all sets A, B, andC, if A ⊆ B then A∪C ⊆ B∪C

Answer with explanation:

We are asked to prove the statement,

For all sets A, B, and C, if A ⊆ B then A∪C ⊆ B∪C

Let us consider set A as:

A={1,3,4,5}

and B={1,2,3,4,5,6,7}

Clearly we may observe that A is a subset of B.

(Since, all the elements of set A are contained in set B.

Hence, A is a subset of B)

Now let us consider set C as:

C={1,2}

Hence,

A∪C={1,2,3,4,5}

and

B∪C={1,2,3,4,5,6,7}

Still we observe that:

A∪C ⊆ B∪C

Since all the elements of the set A∪C are contained in the set B∪C.