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Math Help - Cartesian Product

  1. #1
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    Cartesian Product

    Suppose that A and B are sets, and that A X B is the empty set.
    How can you prove that given any set C, either A is a subset of C or B is a subset of C?
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  2. #2
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    Quote Originally Posted by brudman View Post
    Suppose that A and B are sets, and that A X B is the empty set.
    How can you prove that given any set C, either A is a subset of C or B is a subset of C?
    Surely you can can show effort on this one.
    Under what conditions is it ever true that A \times B = \emptyset ~?

    Give it a try!
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  3. #3
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    Wouldn't A and B have to be empty sets themselves?
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  4. #4
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    Quote Originally Posted by brudman View Post
    Wouldn't A and B have to be empty sets themselves?
    Well at least one of the two would have to be \emptyset.
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  5. #5
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    So how can i write a formal proof without using algebra? or do i have to use algebra?
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  6. #6
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    Quote Originally Posted by brudman View Post
    So how can i write a formal proof without using algebra? or do i have to use algebra?
    Just write it up in sentence/paragraph form.
    The empty set is a subset of every set.
    So you are done.
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  7. #7
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    So the proof will go something like this:

    Suppose that A and B are sets, and that A X B is the empty set.
    Let A = ∅ then ∅ B = ∅ by definition, so A is a subset of C.
    Let B = ∅ then A ∅ = ∅ by definition, so B is a subset of C.

    is this good enough?
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