Functions - is this proof satisfactory?

• charmedbeauty
In summary, the conversation discusses the question of whether the composite function g°f being one-one implies that f is also one-one. The participants discuss the definitions of one-to-one functions and provide a proof by contradiction that shows that if g°f is one-one, then f must also be one-one. They also mention that using Venn diagrams can be a helpful approach for proving this statement.
charmedbeauty

Homework Statement

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.

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?

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

hmm I think I don't understand.

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 let's 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)

emailanmol said:
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 let's 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)

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?

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

charmedbeauty said:

Homework Statement

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.

The Attempt at a Solution

By definition of a function every value of A has to be mapped to a value in B. Likewise every value of B has to be mapped 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?
What is the definition of a one-to-one function ?

SammyS said:
What is the definition of a one-to-one function ?

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.

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

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

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

Hello charmed beauty.

The most elegant propf will be by proving that f(x) can't 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

1. What is the purpose of a proof?

A proof is used to demonstrate the validity of a statement or argument. In the context of functions, a proof shows that a function correctly performs its intended task.

2. What makes a proof satisfactory?

A satisfactory proof must be logically sound and convincing. It should clearly explain the reasoning behind each step and provide evidence to support the conclusion.

3. How can one determine if a proof is correct?

A proof is considered correct if it follows the rules of logic and if each step can be verified to be true. It should also align with any given assumptions or axioms.

4. Can a proof be deemed insufficient?

Yes, a proof can be deemed insufficient if it does not fully address the statement or if it contains errors or inconsistencies. In the context of functions, a proof may also be insufficient if it does not cover all possible scenarios or if it lacks clear explanations.

5. Is there a standard format for writing proofs?

There is no standard format for writing proofs, as different fields and types of proofs may have varying styles. However, a good proof should be well-structured and organized, with clearly defined assumptions, statements, and justifications for each step.

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