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Math Help - Representation of set. Cartesian product

  1. #1
    Junior Member Researcher's Avatar
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    Representation of set. Cartesian product

    Hello.
    When we say A*B={(x,y) : x is an element of A, y is an element of B}, we know that:
    if (x,y) is an element of A*B => x is an element of A, y is an element of B

    Is it also true?
    If x is an element of A, y is an element of B => (x,y) is an element of A*B

    Thanks.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Researcher View Post
    Hello.
    When we say A*B={(x,y) : x is an element of A, y is an element of B}, we know that:
    if (x,y) is an element of A*B => x is an element of A, y is an element of B

    Is it also true?
    If x is an element of A, y is an element of B => (x,y) is an element of A*B

    Thanks.
    Yes, by definition. (x,y) \in A \times B if x \in A, y \in B. But as that is a definition the "if" can be read as an "if and only if". The "only if" gives you the direction you need.
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  3. #3
    Junior Member Researcher's Avatar
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    Thanks Swlabr.
    Could you please introduce a source containing it to me?
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Researcher View Post
    Thanks Swlabr.
    Could you please introduce a source containing it to me?
    Any book on elementary set theory should have it, although I can't give you specific example.
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