 # (b) Show that the 'rule' g:Z_6 rightarrow Z_9 defined by f([a]_6)=[4a]_9 is not a well-defined function. Ishaan Booker 2022-07-15 Answered
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?
1. (b) Show that the 'rule' $g:{Z}_{6}\to {Z}_{9}$ defined by $f\left(\left[a{\right]}_{6}\right)=\left[4a{\right]}_{9}$ is not a well-defined function.
2. Define a function $f:N×N\to N$ by $f\left(\left(a,b\right)\right)=gcd\left(a,b\right)$
(a) show that f is not one-to-one
(b) show that f is onto
3. Let 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
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Step 1
2b) I assume the notation $\left[a\right]6=a\phantom{\rule{0.667em}{0ex}}\mathrm{mod}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}6$. 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 $f\left(\left[1\right]6\right)=\left[4\right]9=4$ but on the other side $f\left(\left[7\right]6\right)=\left[28\right]9=1$. So the map is not well defined.
1a) It is not one-to-one see $f\left(6,8\right)=2=f\left(10,12\right)$.
1b) Look at $f\left(n,n\right)=n$ 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 $f=\mathrm{cos}\left(x\right)$ clearly not onto. And let $g=x$, (the constant function) then $g\circ f$ is onto but f isn't.

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