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Confusion on anti-symmetric and symmetric

  1. Aug 20, 2010 #1
    confusion on "anti-symmetric" and "symmetric"

    Hi guys,
    I am a physics sophomore at next term, recently I am doing a reading on Naive Set Theory on my own. However, I got a few confusion.

    The books said that if A is a subset of B and B is a subset of A, then A=B, but this set inclusion is anti-symmetric,

    on the other hand, based on the axiom of extension. Two sets are equal iff they have the same elements, then. if A=B, then it is symmetric.

    My question is:

    a.) What meant by being symmetric & anti-symmetric
    b.) what is the difference between the two approaches to "A=B"?

    Thank you so much for reading :D
    have a good day.
     
  2. jcsd
  3. Aug 20, 2010 #2

    Landau

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    Re: confusion on "anti-symmetric" and "symmetric"

    The terms 'symmetric' and 'anti-symmetric' apply to a binary relation R:

    R symmetric means: if aRb then bRa.
    R anti-symmetric means: if aRb and bRa, then a=b.

    Thus the relation [itex]\subseteq[/itex] is anti-symmetric: if [itex]A\subseteq B[/itex] and [itex]B\subseteq A[/itex] then A=B.
    The relation "=" is symmetric: if A=B then B=A.
     
  4. Aug 21, 2010 #3
    Re: confusion on "anti-symmetric" and "symmetric"

    thank you so much!
    I can see the difference now.

    would you mind elaborating please? :D
     
  5. Aug 21, 2010 #4

    Landau

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    Re: confusion on "anti-symmetric" and "symmetric"

    Elaborate on what?
     
  6. Aug 21, 2010 #5
    Re: confusion on "anti-symmetric" and "symmetric"

    There is no `other hand' and there are not two approaches to A = B. The axiom of extension is the key principle for set identity: two sets are identical iff they have the same members.

    Now, if it is assumed that A is a subset of B and B is a subset of A, we can prove that they are identical, using this principle.

    For if A is a subset of B then every member of A is a member of B. And if B is a subset of A, then every member of B is a member of A. Thus x is a member of A if and only if it is a member B. Thus A and B have the same members. Thus, by our principle, they are the same set.
     
  7. Aug 25, 2010 #6
    Re: confusion on "anti-symmetric" and "symmetric"

    Thanks for answering, I start to understand it.
    but I am still confused by what practical difference between symmetric and anti-symmetric is?
    in this case(set), they produce same result to me (except the "path")
     
  8. Aug 25, 2010 #7
    Re: confusion on "anti-symmetric" and "symmetric"

    In a symmetric relation, if a is related to b, then b must also be related to a (as happens, for example, in equality). If the relation is antisymmetric, then if a and b are both related to each other, they must be identical (as is the [itex]\leq[/itex] relation). In fact, antisymmetrical relations usually express some kind of weak ordering.

    Picture as a directed graph: in a symmetric relation, if there is an arc between two distinct nodes, then there must be another arc in the opposite direction; for antisymmetry, this can only happen if the nodes are identical.
     
  9. Aug 25, 2010 #8
    Re: confusion on "anti-symmetric" and "symmetric"

    Informally: Your boss can fire you, but you can't fire your boss. You are not equal.
     
  10. Aug 26, 2010 #9
    Re: confusion on "anti-symmetric" and "symmetric"

    so is picking up

    boys pick up girls,
    girls never pick up guys.

    therefore, it is no equality of male and female over the anti-symmetric relation "picking up"
    right? lol

    (well, that's true at least in Asia lol)
     
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