Binary relations and functions

Set

{a,b,c} = {c,a,b} order does not matter

Ordered pair

Two elements where order matter, distinctive roles. (1,3) ≠ (3,1)

Cartesian product A x B

Every possible combination of the ordered pairs

The set of all possible ordered pairs with the first element from A and the second, B.

If A = {1,2} and B = {a,b}, A x B = {(1,a),(1,b),(2,a),(2,b)}

Binary relation

Which elements are connected

Function

A binary relation for which every input has exactly one output. Two inputs can share the same output
— how one quantity changes when another changes (i.e. how disposable income affects consumption)

A relation that is not a function

One input has more than one output or none.
G = {(1,a)(1,b)(2,a)} is not a function, input 1 has two outputs: a and b

Domain

Set of all permitted inputs

For f={(1,3),(2,1),(3,4),(4,6)} the domain is {1,2,3,4} (x coordinate)

Range

Set of all outputs (y coordinate)

Steps to the inverse of a function

1. y = f(x)

2. switch x and y

3. rearrange to get y by itself

4. replace y with f^-1(x)