📝Set theory: functions

function: $f : A \to B$ is a function iff $\forall x \in A. \exists! y \in B \ni f(x) = y$ (i.e., it must be total)

• $A$ is domain

• $B$ is codomain

• Image of function $f$ is a set of possible values (y’s)

  • $y \in Image(f) \iff \exists x \in A \ni f(x) = y$

injective function: $f : A \to B$ is injective iff $\forall x_1, x_2 \in A. f(x) = f(y) \implies x = y$

• no two x’s map to the same y

surjective function: $f : A \to B$ is surjective iff $\forall y \in B. \exists x \in A \ni f(x) = y$

• for all y, there is an x

bijective functions: is a function that is both injective and surjective

• bijective functions have an inverse