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Posers and Puzzles

Posers and Puzzles

  1. Standard member genius
    Wayward Soul
    12 Oct '06 10:22 / 2 edits
    symmetry on an operation R over Z is defined thus,

    xRy => yRx

    e.g. if x=y then y=x, if 3|(x-y) then 3|(y-x)

    anti-symmetry on R over Z is defined thus,

    if xRy and yRx then x=y,

    e.g. x(< =)y and y(< =)x then x=y (< = is less than or equal to)

    Question: is the operation ' < ' symmetric, anti symmetric, neither or both?
  2. 12 Oct '06 16:56 / 1 edit
    [Blank posting]
  3. 12 Oct '06 17:41
    Originally posted by genius
    symmetry on an operation R over Z is defined thus,

    xRy => yRx

    e.g. if x=y then y=x, if 3|(x-y) then 3|(y-x)

    anti-symmetry on R over Z is defined thus,

    if xRy and yRx then x=y,

    e.g. x(< =)y and y(< =)x then x=y (< = is less than or equal to)

    Question: is the operation ' < ' symmetric, anti symmetric, neither or both?
    Maybe relation is more appropriate than operation.
    Depends what meaning you impose on that symbol > but if we assume the default it is neither symmetric, nor anti-symmetric.
  4. 12 Oct '06 18:30 / 2 edits
    Originally posted by ilywrin
    Maybe relation is more appropriate than operation.
    Depends what meaning you impose on that symbol > but if we assume the default it is neither symmetric, nor anti-symmetric.
    not sure, but it seems like it is anti-symmetric to me, because a>b and b>a cannot happen both, and therefore the statement: if a>b and b>a , then a=b is true
  5. 12 Oct '06 19:05
    you're discussing relations and antisymetric is if xRY then NOT yRx as a relation can be either reflexive (for all x xRx) or anti-reflexive (for all x NOT xRx)
  6. 12 Oct '06 22:36
    Originally posted by aginis
    you're discussing relations and antisymetric is if xRY then NOT yRx as a relation can be either reflexive (for all x xRx) or anti-reflexive (for all x NOT xRx)
    no no, antisymmetric really is this by definition: if xRy and yRx, then x=y.
    It is not the opposite of symmetric
  7. 13 Oct '06 09:17
    Originally posted by Skinn13
    no no, antisymmetric really is this by definition: if xRy and yRx, then x=y.
    It is not the opposite of symmetric
    wooops righ you are ...look what a good nights sleep will do
    (xRy ^ yRx)-> x=y looks more familiar. I was thinking antisymmetric AND antireflexive shame on me.
  8. Standard member genius
    Wayward Soul
    13 Oct '06 15:02
    Originally posted by Skinn13
    not sure, but it seems like it is anti-symmetric to me, because a>b and b>a cannot happen both, and therefore the statement: if a>b and b>a , then a=b is true
    yes-i meant relation not operation

    i think you've got it, but could you give a better explanation?
  9. 14 Oct '06 12:24
    Originally posted by genius
    yes-i meant relation not operation

    i think you've got it, but could you give a better explanation?
    what explanation is better than simple logic: (if A , then B) is always true when A is never true
  10. 14 Oct '06 14:33 / 2 edits
    The relation is not symmetric, and it is anti-symmetric.