Question

Prove if A is a subset of B iff B compliment is a subset of A compliment

Answer 1

1) Assume that A ⊂ B.

Choose an element c ∈ B' (B' means complement of B)

Proof by contradiction:

Assume that c ∉ A'

Since c ∉ A' then c ∈ A (that is it is is not an element of A')

Then since A ⊂ B, c ∈ B., This is a contradiction of the assumption that c ∈ B'.

Therefore c ∈ B' implies that c ∈ A'
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2) This is a corollary (implied result) of (1)

Assume that B' ⊂ A'

Call B' the subset D. Call A' the subset E

Then we are trying to prove that since D ⊂ E, then E' ⊂ D'.

But we have already proved this with different letters A and B, so we already know this result.

End proof.

Answer 2

\[A \subseteq B \iff B^{C} \subseteq A^{C}\]

Basically, I just don't know how to do it in a way that isnt really sloppy or in a way thats convincing.....

Are you serious @shamil98?

Answer 3

What?