[SOLVED] one-to-one and onto functions

Feb 2009
5
0
Give an example of two finite sets A and B and two functions f:A goes to B and g:B goes to A, such that f is one-to-one but not onto and g is onto but not one-to-one.
Maybe I'm over thinking this (or under thinking) but I can't seem to find a way that this is even possible. Help please!
 
May 2008
1,024
409
Baltimore, MD (USA)
Give an example of two finite sets A and B and two functions f:A goes to B and g:B goes to A, such that f is one-to-one but not onto and g is onto but not one-to-one.
Maybe I'm over thinking this (or under thinking) but I can't seem to find a way that this is even possible. Help please!
Should be easy. For example, take \(\displaystyle A=\{0\}, B=\{0,1\}\) and define \(\displaystyle f\colon A\to B\) and \(\displaystyle g\colon B\to A\) as follows.

\(\displaystyle \begin{tabular}{c|c}
$A$ & $0$\\\hline
$f(A)$ & $0$
\end{tabular}\)

\(\displaystyle \begin{tabular}{c|cc}
$B$ & $0$ & $1$\\\hline
$g(B)$ & $0$ & $0$
\end{tabular}\)
 
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Feb 2009
5
0
I can't believe I didn't see this to begin with I was really over thinking it! Thanks so much!