(A)symmetries of a function
Functions hold intrinsically some fundamental asymmetries. And these are associated precisely to the injectivity and surjectivity of these functions. This is a cheatsheet to recall the intuition behind and gather some important results which would be helpful to remember eternally.
Left Cancellation, injectivity and left inverse.
Right cancellation, surjectivity and right inverse.
$g\circ f$ injective (surjective) $\implies$ $f (g)$ is injective (surjective).
$y \in f(S) \iff y =f(x)$ for some $x \in S.$
$f(x) \in U \iff x \in f^{-1}(U).$
$f^{-1}(f(S))$ and $S$.
$f(f^{-1}(U))$ and $U$.
$f(A \cap B)$ and $f(A \cup B).$
$f^{-1}\left( U \cup V \right)$ and $f^{-1}\left( U \cap V \right).$