📝Set theory: functions

function: $f : A \to B$ $f : A \to B$ is a function iff $\forall x \in A. \exists! y \in B \ni f(x) = y$ $\forall x \in A .\exists! y \in B ∋ f (x) = y$ (i.e., it must be total)
- $A$ is domain
- $B$ is codomain
- Image of function $f$ $f$ is a set of possible values (y’s)
  - $y \in Image(f) \iff \exists x \in A \ni f(x) = y$
injective function: $f : A \to B$ $f : A \to B$ is injective iff $\forall x_1, x_2 \in A. f(x) = f(y) \implies x = y$ $\forall x_{1}, x_{2} \in A . f (x) = f (y) ⟹ x = y$
- no two x’s map to the same y
surjective function: $f : A \to B$ $f : A \to B$ is surjective iff $\forall y \in B. \exists x \in A \ni f(x) = y$ $\forall y \in B .\exists x \in A ∋ f (x) = y$
- for all y, there is an x
bijective functions: is a function that is both injective and surjective
- bijective functions have an inverse

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