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Math Help - antisymmetric

  1. #1
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    hi how can i tell if if something is anti symmetric i know the definition but its confusing can some one give me a good example.

    The definition goes if (a,b) R implies (b,a) R then a=b
    i don't understand how that would work wouldn't it just mean if it were reflexive and there isn't an (b,a) that relates to (a,b) ? that what the di graph looks like can some one clarify ?


    ohhh as well does being reflexive guarantee transitivity ?
    Last edited by mr fantastic; March 13th 2009 at 05:41 PM. Reason: Merged posts
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  2. #2
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    Antisymmetric relation

    Hello mikeqwerty
    Quote Originally Posted by mikeqwerty View Post
    hi how can i tell if if something is anti symmetric i know the definition but its confusing can some one give me a good example.

    The definition goes if (a,b) R implies (b,a) R then a=b
    i don't understand how that would work wouldn't it just mean if it were reflexive and there isn't an (b,a) that relates to (a,b) ? that what the di graph looks like can some one clarify ?
    The definition simply means that the only way you can have aRb and bRa at the same time in an antisymmetric relation is if a and b are one and the same element. Don't confuse this with a reflexive relation which says that aRa for all elements a.

    There are some pretty straightforward examples here: Antisymmetric relation - Wikipedia, the free encyclopedia. If you don't understand them, tell us where you're having trouble, and we'll take it from there.

    As far as the digraph of an antisymmetric relation is concerned, if a and b are two distinct points (nodes) on the digraph, and there's a directed link from a to b, then there can't be a link back from b to a. If there were a link back, then you would have a pair of elements a and b for which aRb and bRa without a being equal to b.

    Grandad

    PS The answer to your second post: No.
    Last edited by Grandad; March 13th 2009 at 10:19 AM. Reason: Add PS
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  3. #3
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    Quote Originally Posted by Grandad View Post
    Hello mikeqwertyThe definition simply means that the only way you can have aRb and bRa at the same time in an antisymmetric relation is if a and b are one and the same element. Don't confuse this with a reflexive relation which says that aRa for all elements a.

    There are some pretty straightforward examples here: Antisymmetric relation - Wikipedia, the free encyclopedia. If you don't understand them, tell us where you're having trouble, and we'll take it from there.

    As far as the digraph of an antisymmetric relation is concerned, if a and b are two distinct points (nodes) on the digraph, and there's a directed link from a to b, then there can't be a link back from b to a. If there were a link back, then you would have a pair of elements a and b for which aRb and bRa without a being equal to b.

    Grandad

    PS The answer to your second post: No.
    I think i know what your getting at with the digraph
    its the definition that is confusing using the same variables.. a and b can we use some numbers as an example or other letter to clarify , this is what i am thinking
    {(1,1),(1,2)} This is anti symmetric
    {(1,1),(1,2),(2,1)}This is not according to the digraph definition
    {(1,2),(2,1)} this is also not
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  4. #4
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    Antisymmetric relation

    Hello mikeqwerty
    Quote Originally Posted by mikeqwerty View Post
    I think i know what your getting at with the digraph
    its the definition that is confusing using the same variables.. a and b can we use some numbers as an example or other letter to clarify , this is what i am thinking
    {(1,1),(1,2)} This is anti symmetric
    {(1,1),(1,2),(2,1)}This is not according to the digraph definition
    {(1,2),(2,1)} this is also not
    You are right in your examples, but they don't give you much of a feel for what is happening. Let me give you one or two more examples, that might help you to understand this better.

    Consider the set \mathbb{N}, the natural numbers: 1, 2, 3, ...; and suppose that R is the relation 'is a factor of' defined on these numbers. Here are some related numbers, and the ordered pairs that they form:

    • 3 is a factor of 12; so 3R12 and (3, 12) is an element of R
    • 6 is a factor of 12; so 6R12 and (6, 12) is an element of R
    • 12 is a factor of 12; so 12R12 and (12, 12) is an element of R

    ... and so on: if a number a is a factor of a number b, then aRb and the ordered pair (a, b) is an element of the relation R.

    Now suppose that you are told that someone has found a pair of numbers, a and b, such that (a, b) is in the relation, and (b, a) is also in the relation; in other words aRb and bRa. What conclusion can you draw? Well, aRb means that:

    • a is a factor of b

    and bRa means that

    • b is a factor of a

    So how can it be that a divides exactly into b without leaving a remainder, and b also divides exactly into a? The answer is, of course, that a = b. That's the only way in which two numbers can be factors of each other: if they are equal!

    So, for this relation R = 'is a factor of', if aRb and, at the same time, bRa, then a = b. This is what defines an antisymmetric relation.

    A second example is the relation 'is greater than or equal to', simply denoted by the symbol \ge. Again, the only way that a \ge b and b \ge a can be both true at the same time is if a = b. Once again, then, if aRb and bRa, then a = b. So \ge is another example of an antisymmetric relation.

    Does that make it any clearer?

    Grandad
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  5. #5
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    OMG!!!!!!!!!! perfect example!! you should write a book!!! XD my text books are bleh thanks grandad XD
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