# Counterexample for set identity

1. Jun 20, 2011

### WannabeNewton

1. The problem statement, all variables and given/known data
Consider the function $$f:X \to Y$$. Suppose that A and B are subsets of X. Decide whether the following statements are necessarily true (I am including just the one I had trouble with):
(a) if $$A\cap B = \emptyset$$, then $$f[A]\cap f = \emptyset$$

2. Relevant equations

3. The attempt at a solution
I know this statement is false and I know I have to use a counter example. The problem is I am not at all good with counter examples. Could I use a discrete situation as a way of disproving the statement? For example, if I let $$X = \mathbb{R}$$, A = {-1, -2, -3}, B = {1, 2, 3}, and f = {(x,y): y = x^2} then $$A\cap B = \emptyset$$ but f[A] = f = {1, 4, 9} so $$f[a]\cap f\neq \emptyset$$. I don't know if this suffices as a counter example because it is a very specific example so I was hoping you guys could help me come up with one that would be credible?

Last edited: Jun 20, 2011
2. Jun 20, 2011

### Dick

There's nothing wrong with a counterexample being specific. That looks fine to me.

3. Jun 20, 2011

### WannabeNewton

Oh ok. Thank you.

4. Jun 20, 2011

### micromass

Staff Emeritus
The LaTeX symbol for the emptyset is simply \emptyset

5. Jun 20, 2011

### WannabeNewton

Fixed it thanks mate.