Functions

To define a mathematical object like a function, we must first understand what a set is.

A set is an unordered collection of objects like S = {-4, 4, -3, 3, -2, 2, -1, 1, 0}. If a set S is not infinite, we use |S| to denote the number of elements, which is known as the Cardinality of the set. If A and B are finite sets, then |A⇥B|=|A|⇥|B|, which is known as the Cartesian product.

For each input element in a set A, a function assigns a single output element from another set B. A is called the domain of the function, and B, the codomain. A function is a set of pairs (x, y), with none of these pairs having the same first element.

Example: The function with domain {1, 2, 3, . . .}, which doubles its input is the set {(1,2),(2,4),(3,6),(4,8),...}

Example: The function with domain {1, 2, 3, . . .} ⇥ {1, 2, 3, . . .}, which multiplies the numbers forming its input is {((1,1),1),((1,2),2)),...,((2,1),2),((2,2),4),((2,3),6),... ((3,1),3),((3,2),6),((3,3),9),...

The output of a given input is known as the image of that input. The image of q under a function f is denoted by f (q). If f(q)=s, we say q maps to s under f. We write this as q->s. The set from which all the outputs are chosen is a codomain.

We write this as f: D -> F when we want to say that f is a function with domain D and codomain F.