Prove or disprove the following statement: if f is an injective function from A to B, and g is a surjective function from B to C, then gof is a bijective function from A to C.
Really need help. I'm so lost.
Consider $\displaystyle A = \left\{ {1,2,3} \right\}\;,\;B = \left\{ {a,b,c,d} \right\}\;\& \;C = \left\{ {x,y} \right\}$.
Define functions: $\displaystyle f:\left\{ {(1,a),(2,b),(3,c)} \right\}\;\& \;g:\left\{ {(a,x),(b,x),(c,y),(d,y)} \right\}$.
Use those to think about this problem.
You might find Chapter 3 of Part I of Topology and the Language of Mathematics useful. It provides a number of example of problems like yours, with solutions.
A free download of the first 50 pages (which includes Chapter 3 of Part I) is available here:
Bobo Strategy - Topology
Hope it's useful -
Chris
Making the second of your maps the constant map while the first is a non-surjective injection will provide quite a neat solution.
For instance, $\displaystyle \phi: \mathbb{N} \rightarrow \mathbb{N}$, $\displaystyle x \mapsto x^2$. Then, taking the constant map, $\displaystyle \theta: \mathbb{N} \rightarrow \mathbb{N}$, $\displaystyle x \mapsto x$.
Here, the composition of $\displaystyle \phi$ and $\displaystyle \theta$ is just the first map, \phi. Clearly it is injective but not surjective.