Greatest Hits on Quiz 3

quiz3.pdf

Problem 1

Problem 2

Problem 3

Proof of 3b)

To show: If g\circ f=\iota_A, then f is one-to-one.

Proof: Let g\circ f=\iota_A. To show: f is one-to-one, which is equivalent to \forall x,y\in A, f(x)=f(y) \rightarrow x=y. (Not to be confused with \forall x,y\in A, x=y \rightarrow f(x)=f(y), which is true for any function.)

So let x,y\in A and let f(x)=f(y). Then obviously g(f(x))=g(f(y)), but by g\circ f=\iota_A, that means \iota_A(x)=\iota_A(y), which in turn means x=y, which is what we needed to show.