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Surjective, injective and predicate

  1. Oct 23, 2008 #1
    1. The problem statement, all variables and given/known data

    How do I check if my function is surjective?

    How do I check if my function is injective?

    Suppose my function is a predicate and hence characteristic function of some set. How do I determine such a set?

    2. Relevant equations

    Does anyone know to write "The function f: A->B is not surjective? and The function f:A-> B is not injective?" in SYMBOLS using quantifiers and operators.

    3. The attempt at a solution

    If I have two finite sets, and a function between them. I can compute the value of the function at each point of its domain, I can count and compare sets elements, but I don't know how to do anything else.
    I need detailed, explicit instructions for answering the questions IN WORDS. Can anyone help me to solve this problem.

    Cheers mate
    Last edited: Oct 23, 2008
  2. jcsd
  3. Oct 24, 2008 #2


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    Science Advisor

    By showing that the definitions of "surjective" and "injective" hold, of course. For example, to show that a function, f, from A to B, is surjective, you must show that, if y is any member of B, then there exist x in A so that f(x)= y. To show that a function, f, from A to B, is injective, you must show that if f(x1)= y and f(x2)= y, where x1 and x2 are members of A and y is a member of B, then x1= x2.

    Sorry, I don't recognize the term "predicate" as applied to functions.

    I would say that "f is not surjective" means "There exist y in B such that, for all x in X, it is not true that f(x)= y" and "f is not injective" means "for all y in B, if there exist x1 in X such that f(x1)= y and there exist x2 in X such that f(x2)= y, then x1= x2."

    You need to know the DEFINITIONS of "injective" and "surjective"!
  4. Oct 24, 2008 #3
    I was wondering, if f: R -> R is surjective and g: R -> R is surjective, then is f + g also surjective? Intuitively the answer seems to be yes, and I can't think of any counterexamples. Perhaps I'm not thinking hard enough.
  5. Oct 24, 2008 #4
    f(x)=x is surjective (and injective), g(x)=-x is also surjective (and injective), what is the sum of f and g?
  6. Oct 24, 2008 #5
    Ah, okay, thanks. These questions come up so often in my theoretical calc class and they often amount to using the additive and multiplicative inverses for counter examples. I need to keep that in mind.
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