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Math Help - Does this proof work?

  1. #1
    Senior Member Danneedshelp's Avatar
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    Does this proof work?

    Q:Prove that P(A∩B)=P(A)∩P(B).

    A:Let x∈P(A∩B) be arbitrary ⇒ x⊆A∩B
    ⇔ ∀y(y∈x⇒y∈(A∩B))
    ⇔ ∀y(y∈x⇒y∈A and y∈B)
    ⇔ x⊆A and x⊆B
    ⇔ x∈P(A) and x∈P(B)
    ⇔ x∈P(A)∩P(B) ■

    Does this work? I'm not sure if I have used the logical connectives correctly, such as breaking up the subset definition the way I did. I'm just trying prove things using iff statements for the practice.

    Thanks.
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  2. #2
    MHF Contributor

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    It works, but has more steps than necessary.
    But it does work.
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