Question

Prove A subset B if and only if B complement subset of A complement

Answer 1

Must show (1) If A is a subset of B, then B complement is a subset of A complement, and (2) if B complement is a subset of A complement, then A is a subset of B.

We show (1). Part (2) is similar. -- Let A' and B' represent A complement and B complement, respectively.

Suppose A is a subset of B and x is an element of B'. (If B' has no elements, then B' is the empty set, and is by def a subset of every set). So x is not in B. But A is a subset of B. So x is not in A. So x IS in A'. Therefore x in B' implies x in A'. Therefore by def. B' is a subset of A'.

I hope this helps.
dj

Answer 2

thank you very much :D