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Math Help - Ordered Pair

  1. #1
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    Ordered Pair

    A definition in my Abstract Algebra class on an ordered pair is as follows:

    Let A and B be sets. Given a is an element of A and b is an element of B, the ordered pair (a,b) is the set {{a},{a,b}}.


    Why isn't it {{a}, {a,b}, {b}} ? Or just {a,b}?

    I guess I don't understand why {a} is included.

    Any help would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by kid funky fried View Post
    A definition in my Abstract Algebra class on an ordered pair is as follows:

    Let A and B be sets. Given a is an element of A and b is an element of B, the ordered pair (a,b) is the set {{a},{a,b}}.


    Why isn't it {{a}, {a,b}, {b}} ? Or just {a,b}?

    I guess I don't understand why {a} is included.

    Any help would be greatly appreciated.
    The first reason why do we do it at all is that we want the notion of ordered pair to be defined as a set. We want to encode somehow the property of the ordered pair (a,b) that a comes first and b comes second - this is how we understand what " (a,b) is an ordered pair" means. And this is what \{a,b\} and \{\{a\}, \{a,b\}, \{b\}\} fail to provide ( \{a,b\} doesn't tell us that a comes first, it is just a set containing elements a and b, \{a,b\}=\{b,a\})
    The definition \{\{a\}, \{a,b\}\} of the ordered pair (a,b) is cleverly chosen. If we want to say " x is on the first position in (a,b)" in set-theoretical language, we just say " x is an element of all elements of \{\{a\}, \{a,b\}\}". If we want to say " x is on the second position in (a,b)" , we say "there exists an element of \{\{a\}, \{a,b\}\} that contains x AND it is not true that there are two distinct elements of \{\{a\}, \{a,b\}\} both containing x".
    Moreover, the characteristic property of oredered pairs, " (a,b)=(c,d) if and only if a=c and b=d" remains true if we use this definition: " \{\{a\}, \{a,b\}\}=\{\{c\}, \{c,d\}\} if and only if a=c and b=d" (try to verify it).
    This definition comes from Kuratowski and is not the only possible one, but people like it for its properties and have chosen it.
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  3. #3
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    Thanks

    Thank you. I appreciate you taking the time to give such a thoughtful response.
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