# Proving a function f is one to one if and only if \$f(Abigcap B) = f(A) bigcap f(B)\$ Code Answer

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Prove that if \$f:Xrightarrow Y\$ is a function then \$f(Acap B) = f(A)cap f(B)\$ for all subsets \$A\$ and \$B\$ if and only if \$f\$ is one to one.

Let \$x,yin Abigcap B\$ where \$Abigcap B\$ is not empty and \$f(x) = f(y)\$

\$\$xin Acap B rightarrow f(x)in f(Acap B) rightarrow f(x)in f(A)cap f(B)\$\$

\$\$yin Acap B rightarrow f(y)in f(Acap B) rightarrow f(y)in f(A)cap f(B)\$\$

Since \$f(x) = f(y)\$ then \$f\$ is one to one if and only if \$f(Acap B) = f(A)cap f(B)\$.

Not sure if this is right at all, if anyone can do a proof of this similar to what I have I would greatly appreciate it.