Proving Injectivity: f ∘ g = f ∘ h ⇒ g = h

[rallycar18]

Prove that:

If f : X → Y is injective, g, h : W → X, and f ∘ g = f ∘ h, then g = h.

 

[Tedjn]

What is your progress on this problem?

 

[rallycar18]

What is your progress on this problem?

Definition of injective for If f : X → Y :
For all y $$\in$$ Y, there exists at most one x $$\in$$ X such that f(x) = y

Because f : X → Y and g, h : W → X,

f ∘ g : W → X → Y and f ∘ h : W → X → Y

so f ∘ g, f ∘ h : W → Y


that's where I get stuck.

 

[lanedance]

note that an injective function is a function that preserves distinctness... so you can consider the inverse of f

 

[rallycar18]

note that an injective function is a function that preserves distinctness... so you can consider the inverse of f

Then f^(-1) : Y → X is also injective.. but I don't see what I can do from that.

 

[Tedjn]

It may be easier to see via contradiction. If there is w in W such that g(w) is not equal to h(w), what happens to (f ∘ g)(w) and (f ∘ h)(w)?

 

[lanedance]

or if f o g = f o h, then apply f-1