Let $A, B, C$ be nonempty sets with total order. and let $f\colon A \rightarrow B$ and $g\colon B \rightarrow C$ be maps. Prove these statements:
a) If $f, g$ are antitone, then $g \circ f$ is isotone
b) If $f, g$ are strictly isotone, then $g \circ f$ is injective
c) If $f, g$ are injective, then $g \circ f$ strictly is isotone
I am stuck now not knowing where and with what to start. I started like this:
$f\text{ is antitone} \quad:\Longleftrightarrow\quad x
$g\text{ is antitone} \quad:\Longleftrightarrow\quad x
... I don't know how to show that $g \circ f$ is isotone. :(
Can someone help me please with all these?
Thanks a lot.