Chapter 5.4 functions

Functions lecture yay

Functions are sets

Functions are a subset of relations

Definition
f(S) = T

S is the set of inputs

T is the set of inputs

A function is a set of ordered pairs {S, T} the first coordinate S cannot appear more than once

S restrictions
Often there are restrictions on input.

Equality of functions

 * same domain
 * same codomain
 * f(a) = g(a) for all in their domain

Piecewise functions
They do not need to have a specific formula, can change based on place. Piecewise functions do this

quus is a famous piecewise function used by Kripke in the rule following paradox

One to One
One to one functions have no overlap in codomain to domain

Same as surjective

-Jectivities

 * surjective - all elements in the codomain have an element in the domain that goes to them
 * Every odd degree polynomial is surjective
 * Covers entire domain, like surveillance
 * injective - one to one - injects to one value
 * bijective - both

Function composition
Composition is just nested functions