The proof of this is very, very simple.
Hi -
A proof using elements will look something like this:
First, a reminder of the definition of a subset:
- A set P is a subset of a set Q if and only if every element of P is also an element of Q.
We need to prove A is a subset of B iff B' is a subset of A'.
- So let's assume first that A is a subset of B, and prove that B' is a subset of A'.
Choose an element x in B'. This means that x is not in B. In this case x can't be in A either, because, by definition of a subset if it were in A, then it would be in B. Now if x isn't an element of A, it is an element of A'.
We have just shown that whenever we choose an element x of B', it is also an element of A'. Therefore B' is a subset of A'.
This concludes the first half of the proof.
Now for the second half. We begin by saying:
- Let's assume that B' is a subset of A'; we now need to prove that A is a subset of B.
Choose an element x in A. This means that it's not in A'. So it can't be in B' either, because if it were it would be in A' (again because of the definition of a subset). Now if x isn't in B', it must be in B.
We have just shown that whenever we choose an element x of A, it is also an element of B. Therefore A is a subset of B.
That concludes the proof.
Hope that helps.
Grandad