Set Theory: A is a subset of B. Prove that C - B is subset of C - A

Question

A ⊆ B. Prove that C - B ⊆ C - A.

My approach:

Take element k ∈ C - B. This means that k ∈ C and k ∉ B. Since k ∉ B then k ∉ A because A ⊆ B. It's also true that k ∈ C - A because k ∈ C and k ∉ A. Therefore C - B ⊆ C - A because k was found in C - A and C - B.

I'm not sure how to convince myself of the last statement "Therefore C - B ...". Recalling the meaning of A ⊆ B, I know every element in A is found in B. I saw that k ∈ C - B and by further reasoning I found k ∈ C - A. Since I've stated that k is an arbitrary element, does that mean that I've proven that any k found in C - B is also found in C - A? Is it enough to say that since I at least found one k in C - B that's also in C - A then I know C - B ⊆ C - A?

Answer 1

go with the arbitrary.

Another way to write the proof:

Using ' for complement, you could also write

C - B = CB' = CB'A' U CB'A

If A ⊆ B, B'A = Ø, and CB'A = Ø, so C - B = CB'A' U CB'A =CB'A' U Ø = CB'A'

C - A = CA' = CB'A' U CBA'

C - B = CB'A' ⊆ CB'A' U CBA' = C - A

C - B ⊆ C - A