Need help with a set theory proof?

Question

Prove: If A is a subset of B, then A union C is a subset of B union C.

My first inclination was to do a direct proof. But all I ended up doing was defining the initial subset.

Let x be any object.
Suppose A subset B.
Then x is an element of A implies x is an element of B.

Where should I go from there?

I also tried to work backward from the consequent, but got lost in the four different combinations of A U C subset B U C.

I don't think contradiction would work, either.

Please, I need suggestions or help getting on the right track.

Thanks

Answer 1

Suppose A⊆B, and let x∈A∪C. Then x∈A ∨ x∈C. If x∈C, then x∈B∪C, and if x∈A, then x∈B (since A⊆B), and so x∈B∪C. In either case, if x∈A∪C then x∈B∪C, so A∪C⊆B∪C. Q.E.D.

Answer 2

Suppose that A is a subset of B and that x is an element of AuC. We want to show that x is also an element of BuC.

If x is an element of AuC then there are two cases to consider: (1) x is an element of A or (2) x is an element of the remainder of AuC.

If (1) is true, then x is also an element of B, since A is contained in B. As a result x must be an element of BuC.

If (2) is true, then x must be an element of C. Since x is an element of C, it must also be an element of BuC.

So we have shown that if x is an element of either of the two partitions of AuC it will be an element of BuC, as desired.