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.