📝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

Backlinks

• 📝 Category Theory: Epimorphism
• 📝 Category Theory: Monomorphism
• 📝 § Category Theory
• 📝 § Math