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 September 29th, 2010, 04:29 AM #1 Newbie   Joined: Sep 2010 Posts: 18 Thanks: 0 math analysis Helllo every body this is my first questions in this forum and i expect to help ^^ Let f be a one-to-one function from A into B with B finite show that A is finite Let f be a one-to-one function from A into B with B countable show that A is countable
 September 29th, 2010, 05:49 AM #2 Senior Member   Joined: Feb 2009 Posts: 172 Thanks: 5 Re: math analysis $f:A\rightarrow B$, $B$ finite. Since $f(A)\subset B$ and $B$ is finite we have that $f(A)$ is finite. Consider the function $g:A\rightarrow f(A)$ defined as $g(a)=f(a)$ for all $a\in A$. We have that $g$ is an injection since $f$ is an injection and we have that $g$ is surjective since for all $y\in f(A)$ there exists $a\in A$ such that $f(a)=g(a)=y$. Hence $g$ is a bijection from $A$ to $f(A)$ and then since $f(A)$ is finite $A$ is finite. $f:A\rightarrow B$, $B$ countable. Since $f(A)\subset B$ and $B$ is countable we have that $f(A)$ is countable. Consider the function $g:A\rightarrow f(A)$ defined as $g(a)=f(a)$ for all $a\in A$. We have that $g$ is an injection since $f$ is an injection and we have that $g$ is surjective since for all $y\in f(A)$ there exists $a\in A$ such that $f(a)=g(a)=y$. Hence $g$ is a bijection from $A$ to $f(A)$ and then since $f(A)$ is countable $A$ is countable.
 September 29th, 2010, 02:58 PM #3 Newbie   Joined: Sep 2010 Posts: 18 Thanks: 0 Re: math analysis thank you parasio i really appreciate this
 September 29th, 2010, 05:17 PM #4 Senior Member   Joined: Feb 2009 Posts: 172 Thanks: 5 Re: math analysis You're welcome. I hope you like the forum and continue to show up.

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### "Let f be a one-to-one function from A into B with B finite. Show that A is finite."

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