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Math Help - Proof involving sets, crossproducts, and intersections

  1. #1
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    Proof involving sets, crossproducts, and intersections

    I have this problem, i'm not quite sure how to go about proving it. If anyone could point something helpful out or ideas, that would be great.

    Prove: (AxB)∩(CxD)=(A∩C)x(B∩D)

    Thanks.
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Proof involving sets, crossproducts, and intersections

    Prove two inclusions.
    1. (A \times B)  \cap (C \times D)  \subset (A \cap C) \times (B \cap D)
    Let x \in (A \times B)  \cap (C \times D) then  x \in A \times B \wedge x \in C \times D. Let x=(u,v) then we have (u,v) \in A \times B \wedge (u,v) \in C \times D, thus u \in A \cap C and v \in B \cap D. Hence  x = (u,v) \in (A \cap C) \times (B \cap D).

    2. Prove the other inclusion ...
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