Surjective, injective and predicate

[reven]

Homework Statement

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?

Homework 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.

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

 

[HallsofIvy]

Homework Statement

How do I check if my function is surjective?

How do I check if my function is injective?

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.

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

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

Homework 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.

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."

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

You need to know the DEFINITIONS of "injective" and "surjective"!

 

[snipez90]

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.

 

[d_leet]

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.

f(x)=x is surjective (and injective), g(x)=-x is also surjective (and injective), what is the sum of f and g?

 

[snipez90]

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.