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Identity Relation and Function

  1. Dec 4, 2008 #1
    1. The problem statement, all variables and given/known data

    I am reading a book on relations on function and I am very confused with identity relation and function. Any help on understanding I relation and I function will be appreciated.

    2. Relevant equations

    A function from A to B is a relation f from A to B such that
    a) the domain of F is A
    b) if (x,y) [tex]\in[/tex] to F and (x,z) [tex]\in[/tex] f, then y = z

    3. The attempt at a solution

    x [tex]\in[/tex] A , IA(x)= x
  2. jcsd
  3. Dec 4, 2008 #2


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    Homework Helper

    I am very confused by your question. What are f and F?
    A relation f from A to B is a subset of AxB
    A function f is a relation from A to B such that
    a)The domain of f is A.
    b)If (x,y) and (x,z) are elementa of f, then y=z.

    So a relation is a special type of function. All functions are relations, but not all relations are functions.

    Say we have a relation R from A to B.
    R is a subset of AxB.
    This means if we chose x in A and y in B it makes sense to ask questions like
    Is (x,y) in R?
    Is (x,z) in R?
    Is x in the domain of R?
    Any combination of answers is possible, but say we have a relation f from A to B.
    and ask
    1)Is (x,y) in R?
    2)Is (x,z) in R?
    3)Is (x,y)=(x,z)?
    4)Is x (x is in A) in the domain of R?
    4 is always true
    If at least 2 of 1,2,3 are true then they all are

    Say we have a relation marrage from men to women
    we can have
    (Bob,Jill) and (Bob,Beth) in marrage

    We might have Bob is not married

    Say we have a function marrage from men to women
    we cannot have
    (Bob,Jill) and (Bob,Beth) in marrage unless Jill=Beth
    but we can have
    (Bob,Jill) and (Sam,Jill) with Bob!=Sam
    We cannot have Bob is not married.
    We can have Jill is not married.
  4. Dec 4, 2008 #3


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    Staff Emeritus
    Science Advisor

    In other words, don't use "f" and "F" to mean the same thing!

    Perhaps what is confusing you is that a "function" is a special type of "relation".

    Every function is a relation but the other way is not true: the relation {(x,y)|x2+ y2= 1} is not a function.

    The identity function {(x,x)} for all x in the base set is also the identity relation- there is no difference.
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