Assuming that all sets are subsets of a universal set U use

Assuming that all sets are subsets of a universal set U, use an elemental argument to prove that for all sets A B, if A B, then A C B C .

Solution

Suppose: Given that
All set are subset of universal set U
to Proove
If A B then A U C B U C.

Proof:
Let x A U C then x A or x C.
In the case that x A, since A B,
it follows that x B; and hence x B U C.
On the other hand, if x C,
then it follows immediately that x B U C.
In either case, if x A U C, then x B U C.
Therefore, A U C B U C.

Hence proved