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Math Help - Proofs

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    Proofs

    Let p,r,q be sets. Prove that if p is a subset of r and r is a proper subset of q, then p is a proper subset of q
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    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by sderosa518 View Post
    Let p,r,q be sets. Prove that if p is a subset of r and r is a proper subset of q, then p is a proper subset of q
    Consider an element x \in p.

    As p \subset r it follows by definition of a subset that x \in r.

    But as r \subset q, it follows that if x \in r then x \in q.

    So, given that x \in p we have shown that x \in q as well.

    That is all you need to do to show that p \subset q.
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  3. #3
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    But you still need to prove that p is a proper subset of q! That is, that there exist a least one member of q that is not a member of p. To do that use the fact that r is a proper subset of q. Notice that the problem does NOT say that p is a proper subset of r!
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    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    But you still need to prove that p is a proper subset of q! That is, that there exist a least one member of q that is not a member of p. To do that use the fact that r is a proper subset of q. Notice that the problem does NOT say that p is a proper subset of r!
    Good call. Care to finish the proof off then? I confess my brain's had a battering tonight.
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