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Math Help - cartesian product proofs

  1. #1
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    cartesian product proofs

    A X (B U C)= (A X B) U (A X C)

    proof:
    Assume (x,y) is an element of A X (B U C). This means x is an element of A and y is an element of B or y is an element of C. Since (x,y) can be x as an element of A and y as an element of B, (x,y) is an element of A X B. Since (x,y) can also have x as an element of A and y as an element of C, (x,y) is an element of A X C.

    A X (B-C)= (A X B) - (A X C)
    not sure

    A X (B intersect C)= (A X B) intersect (A X C)

    please help with these proofs im not sure how the form of proofs about cartesian products should be
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  2. #2
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    Quote Originally Posted by leinadwerdna View Post
    A X (B-C)= (A X B) - (A X C)
    not sure
    Here is one. They all work about the same way.
    \begin{gathered}<br />
  (x,y) \in A \times (B\backslash C) \hfill \\<br />
  x \in A \wedge y \in B \wedge y \notin C \hfill \\<br />
  \left( {x \in A \wedge y \in B} \right) \wedge \left( {x \in A \wedge y \notin C} \right) \hfill \\<br />
  (x,y) \in A \times B \wedge (x,y) \notin A \times C \hfill \\<br />
  (x,y) \in \left( {A \times B} \right)\backslash \left( {A \times C} \right) \hfill \\ <br />
\end{gathered}
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  3. #3
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    is a slash equal to the difference
    also is the first 1right
    Last edited by leinadwerdna; October 29th 2009 at 10:58 AM. Reason: alsois the first one right
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  4. #4
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    i dont understand
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  5. #5
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    Quote Originally Posted by leinadwerdna View Post
    i dont understand
    Maybe you should try to get some one-on-one help from your instructor.
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  6. #6
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    A X (B U C)= (A X B) U (A X C)
    Proof:
    (x,y) E A X (B U C).
    x E A and y E (B U C)
    X E A and (y E B U y E C)
    (x E A and y E B) or (x E A and y E C)
    (x,y) E A X B or (x,y) E A X C
    (x,y) E (A X B) or (A X C)

    and the argument can be reversed

    the only thing I cant justify is how I go from step 3 to step 4 and how I go from step 4 to step 3.

    E= element of
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  7. #7
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    cartesiian proof

    Proof:
    (x,y) E A X (B intersect C)
    x E A and y E (B intersect C)
    x E A and y E B and y E C
    (x E A and y E B) and (x E A and y E C)
    (x,y) E A X B and (x,y) E A X C
    (x,y) E (A X B) intersect (A X C)

    again i am only not sure how to justify going from step 3 to step 4 and how to go from step 4 to step 3

    the argument reverses to prove they are equal

    E= element of
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