We finished our list of proof techniques, adding implication () and existentials (). We then defined functions and discussed some examples.

Reading: MCS 4.3 — 4.5
This material was covered in lectures 2 (modeling problems) and 4 (functions) in 2017sp.
File:Lec05-board.pdf

Proof techniques continued

TODO

Functions

Function definition

Definition: Image

If , then the image of (written ) is the set of values that are output by . Formally, or more succinctly, .

Function examples

let be given by [math]f(x) := x^2 [/math]. The domain and codomain of [math]f [/math] are both ; the image is .

we often draw functions:

The domain of [math]f [/math] is , and the codomain of [math]f [/math] is .

another way to draw a function is with a table:

x	[math]f(x) [/math]
a	1
b	1
c	1

This almost describes a function; the domain is clearly [math]\{a,b,c\} [/math] (because if there were any other domain elements, [math]f [/math] would not be a function); and the rule is clear. However, the codomain is not clear: is it ? Or ? Or , or ? When describing a function with a table, the codomain should be specified somewhere.

SP18:Lecture 5 Functions

Proof techniques continued

Functions

Function examples