Is the following statement true or false? Provide a proof or counter example.

"For any function $f: X \rightarrow Y$, and any subset $A$ of $X$, we have $$f(A \cap (X\backslash A)) = f(A) \cap f(X\backslash A)$$

I think this is false because how can some element be in a set and also not be in its set. But I'm not sure

  • 2
    $\begingroup$ The point is that $f(A\cap A^c)=f(\emptyset)=\emptyset$, but $f(A)\cap f(A^c)$ does not necessarily need to be empty. $\endgroup$ – JMoravitz Apr 11 '17 at 20:22


$A \cap A^{c} = \emptyset$ so $f(A \cap A^{c}) = \emptyset$ but take $f\colon [-1,1] \to \mathbb{R}, f(x) = |x|$ then $f((-1,0)) \cap f([0,1)) = (0,1) \cap [0,1) = (0,1)$


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