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Math Help - subset and powerset proof by contra

  1. #1
    Member pberardi's Avatar
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    subset and powerset proof by contra

    Where is the contradiction? Also can this be proven directly?

    Prove that if x and y are sets such that P(x) is a subset of P(y) then x is a subset of y.

    Pf. by contradiction

    Assume x is not a subset of y
    Then there exists an element z in x which is not in y
    z is in P(x)

    I am stuck can someone show me the contradiction?
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by pberardi View Post
    Where is the contradiction? Also can this be proven directly?

    Prove that if x and y are sets such that P(x) is a subset of P(y) then x is a subset of y.

    Pf. by contradiction

    Assume x is not a subset of y
    Then there exists an element z in x which is not in y
    z is in P(x)

    I am stuck can someone show me the contradiction?
    Hi pberardi.

    Slight correction: \color{red}\{z\} is in P(x).

    Well, you are given P(x) is a subset of P(y). \therefore\ \{z\}\in P(x)\ \implies\ \{z\}\in P(y)\ \implies\ z\in y. There is your contradiction.
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