# Thread: Proof: A-B⊆C if and only if A-C⊆B.

1. ## Proof: A-B⊆C if and only if A-C⊆B.

Having trouble figuring this proof out. Can anyone help me?

Prove A-B⊆C if and only if A-C⊆B.

2. ## Re: Proof: A-B⊆C if and only if A-C⊆B.

Note that X ⊆ Y is equivalent to X ∩ Y' = ∅ where Y' is the complement of Y and ∅ is the empty set. And, as you already know, X - Y = X ∩ Y'. Using these facts, rewrite both sides.

3. ## Re: Proof: A-B⊆C if and only if A-C⊆B.

You can also do this directly from the definitions. To prove "$\displaystyle X\subseteq Y$" show that "if x is in X then it is in Y".

Here, we want to prove $\displaystyle A- C\subseteq B$ so start "if x is in A- C" and try to conclude that x is in B. If x is in A- C then it is in A but not in C. But we know that $\displaystyle A- B\subseteq C$ so every member of A except those that are also in B. Since x is in A but NOT in C, it is in B.

To prove the other way "if $\displaystyle A- C\subseteq B$ then $\displaystyle A- B\sdubseteq C$ do the opposite "if x is in A- B" and show it must be in C.