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Math Help - Cartesian Products: Proof of Equality of Sets

  1. #1
    gryphon_gold
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    Cartesian Products: Proof of Equality of Sets

    Problem:

    If S x T = U x V, then show that S = U and T = V.

    Incomplete Solution:

    Step 1) S x T = {(a,b) | a e S, b e T}; U x V = {(c,d) | c e U, d e V}

    Step 2) {(a,b) | a e S, b e T} = {(c,d) | c e U, d e V}

    .
    .
    .

    Step n-1) a = c for all a e S, c e U; b = d for all b e T, d e V

    Step n) S = U; T = V

    Question: How do I make a formal connection between steps 2 and N-1?
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  2. #2
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    Quote Originally Posted by gryphon_gold
    Problem:

    If S x T = U x V, then show that S = U and T = V.
    Assume
    S\times T=U\times V and S=U \mbox{ and }T=V is false. Then, be de Morgan laws. S\not =U \mbox{ or } T\not = V
    If,
    S\not = U then \exists [a\in S \mbox{ and }a\not \in U] \mbox{ or }[a\not \in S \mbox{ and }a\in U]
    Then,
    \{(s,t)|a\in S \mbox{ and }t\in T\}\not = \{(u,v)|u \in U\mbox{ and } v\in V\}
    Contradiction,
    Thus,
    S=U \mbox{ and } T=V
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