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Math Help - Ordered pair. need help in understanding.

  1. #1
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    Ordered pair. need help in understanding.

    Need help in understanding this theorem..

    Theorem 2.1.1. Let <a, b> and <c, d> be ordered pairs. Then <a, b> = <c, d> if and only if a = c and b = d.
    Remark. The expression “if and only if” means that
    1. If <a, b> = <c, d> , then a = c and b = d.
    2. If a = c and b = d, then <a, b> = <c, d> . (This is called "the converse" of 1.)
    So, we have to prove two directions, namely 1. and 2. Usually, “if and only if” is abbreviated as simply “iff”.
    Proof. “⇒”: Suppose <a, b> = <c, d> ; then, by definition of ordered pair,
    <a, b> = {{a}, {a, b}},
    <c, d> = {{c}, {c, d}},
    and since they are equal by our hypothesis, we have
    {{a}, {a, b}} = {{c}, {c, d}}.
    We consider two cases:
    1. a = b: Then,
    <a, b> = {{a}, {a, b}} = {{a}} = {{c}, {c, d}},
    hence, {a} = {c} which implies a = c. Furthermore, {a} = {c, d} = {a, d}
    which implies d = a = b. Thus, for this case we have shown that a = c and b = d.

    I got stuck at this d=a=b. how?
    someone please explain.
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  2. #2
    Super Member Gamma's Avatar
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    Quote Originally Posted by chakravarthiponmudi View Post
    Need help in understanding this theorem..

    We consider two cases:
    1. a = b: Then,
    <a, b> = {{a}, {a, b}} = {{a}} = {{c}, {c, d}},
    hence, {a} = {c} which implies a = c. Furthermore, {a} = {c, d} = {a, d}
    which implies d = a = b. Thus, for this case we have shown that a = c and b = d.

    I got stuck at this d=a=b. how?
    someone please explain.
    Well if you are good through {a} = {c, d} = {a, d} you are golden. the set {a} has 1 element, the other two have 2 elements, for finite sets to be equal, the cardinality must be the same. but that means c,d and a,d are really they same thing and they must all actually be a to have the sets be equal. So that gives us a=c=d, but recall which case we are in, for a=b. so we get b=a=c=d, which means that in particular a=c and b=d, they are just all the same too. It is kind of like the trivial case I guess. Does that explain it?
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  3. #3
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    Thank you

    Yes. I got it now.. Thankyou.
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