functions - is this proof satisfactory?

charmedbeauty

1. The problem statement, all variables and given/known data

Let f be a function from A to B and g a function from B to C. Show that if the composite function g°f is one-one, then f is one-one.



2. Relevant equations



3. The attempt at a solution

By definition of a function every value of A has to be maped to a value in B. Likewise every value of B has to be maped to a value in C. So by definition of 1-1... since g°f satisfies this rule and g°f is a composite function... Therefore f must be 1-1.

Does this make a valid argument? and justify the answer?

Oster

Its not so clear to me.
If f(x) = f(y) for some x,y in A you have to show x=y.

charmedbeauty

hmm I think I dont understand.

emailanmol

You have to prove that if f(x) is not one one
then function g(f(x) is not one one.

Suppose that f(x) is not one one and for two values of x lets say x1 and x2(both lying in the domain) give a value y1.

Now this y1 lies on domain of function g.

What happens when you calculate g(y1)?

What does it tell us about our original assumption the f(x) is not one one.

p.s

Your proof is not right.(even if you understand why f should be one one, the proof is not satisfactory)

charmedbeauty

But I have to prove f is one-one.

But from the assumption say I have 3 sets A,B,C the same as the original question.
Now x1 & x2 lie in A. when we plug in the values x1 & x2 into f wwe have they both = y1 which lies in B. So from this picture alone the function f :A→B is not one-one. But we have to prove the opposite!

And to this statement.


"What happens when you calculate g(y1)?"

if the value of f at x1 and x2 = y1.... then the above statement is just saying g(f(x)).

I'm confused?

emailanmol

You are given g(f(x)) is one one.now you are supposed to prove f(x) is one one.


To prove this we say let f(x) be many-one(i.e not one one).


In that case we observe that g(f(x)) comes out to be many one and not one one(How?).However we are told that g(f(x)) is one one.
This contradiction signifies that our assumption of f(x) being many one is wrong and therefore f(x) is one one.

This is the way you can proceed.Solving the indivudual bits is now upto you

SammyS

What is the definition of a one-to-one function ?

charmedbeauty

No 2 elements in X can be maped to the same element in Y. for f:X→Y
ie, f(x1) & f(x2) cannot both = y1.

SammyS

As I read your proof, there is nothing in it which explicitly uses this fact.

charmedbeauty

Can I prove it using a venn diagram?
Im not so good putting what I want to say in words.

emailanmol

Hello charmed beauty.

The most elegant propf will be by proving that f(x) cant be many one(which i mentioned a while ago).

Drawing Venn-Diagrams is also a v useful approach.
However, as the no. of functions forming the composite increase, proving using Ven Diagrams will get v big.

But they are good way to start as they convey a lot of information