I'm doing a review for my discrete math test on functions and I'm having troubles with a few questions. Can I get some guidance in how to do these questions so I can be more prepared for the test? Thanks
(b) Show that the 'rule' $g:Z_6\to Z_9$ defined by $f([a]_6) = [4a]_9$ is not a well-defined function.
Define a function $f: N\times N \to N$ by $f((a,b)) = \gcd(a,b)$
(a) show that $f$ is not one-to-one
(b) show that $f$ is ontoLet $A$, $B$, $C$ be non-empty sets and let $f: A \to B$ and $g: B \to C$ be functions.
(a) Show that it $g\circ f$ is onto, then $g$ is onto
(b) Find an example of functions $f$ and $g$ such that $g\circ f$ is onto but where $f$ is not onto
2b) I assume the notation $[a]6=a\mod6$ (I admit I don't know this notation.) Then the function is not well defended because if it was well defined then it should give the same answer when you take a different representative of the equivalent class. Then note $f([1]6)=[4]9=4$ but on the other side $f([7]6)=[28]9=1 $. So the map is not well defined.
1a) It is not one-to-one see $f(6,8)=2=f(10,12)$.
1b) Look at $f(n,n)=n$ so from this you can conclude that it is onto.
2a) If $g\circ f$ is onto then $g$ is onto on the image of $f$ therefore it is also onto on $B$, thus $g$ is onto.
2b) If you would have $A=B=\mathbb{R}$ and $C={x}$ (just one point) then let $f=\cos(x)$ clearly not onto. And let $g=x$, (the constant function) then $g\circ f$ is onto but $f$ isn't.