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Math Help - theorems of set theory

  1. #1
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    theorems of set theory

    Prove by using elements prove:

    A is a subset of B iff B' is a subset of A'
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  2. #2
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    The proof of this is very, very simple.
    P \Rightarrow Q\,\text{ if and only if } \,\neg Q \Rightarrow \neg P
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  3. #3
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    Proof using elements

    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
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