Proof Wrong: Where f(A∩B) ≠ f(A) ∩ f(B)

[woundedtiger4]

I know that f(A∩B) is not equal to f(A) ∩ f(B) but i don't know that where am i wrong in the following proof...:( can someone please give me an intuitive example?

 

[HallsofIvy]

In order to prove "$X= Y$" you must prove both "$X\subseteq Y$" and $Y\subseteq X$.

You want to prove that $f(A\cap B)$ is not equal to $f(A)\cap f(B)$ and your give proof shows only that $f(A\cap B)\subseteq f(A)\cap f(B)$.

So look for a counter example in which $f(A)\cap f(B)$ is NOT a subset of $f(A\cap B)$.

That is, find a function f and values p and q, p in A, q in B such that f(p)= f(q).

 

[woundedtiger4]

In order to prove "$X= Y$" you must prove both "$X\subseteq Y$" and $Y\subseteq X$.

You want to prove that $f(A\cap B)$ is not equal to $f(A)\cap f(B)$ and your give proof shows only that $f(A\cap B)\subseteq f(A)\cap f(B)$.

So look for a counter example in which $f(A)\cap f(B)$ is NOT a subset of $f(A\cap B)$.

That is, find a function f and values p and q, p in A, q in B such that f(p)= f(q).

Thank you sir.
I understand your point & I can guess the best example is one to many which is not true, & i have tried to show it in proof.