Basics

Saying y is a function of x means y is related to x by some formula

An ordered pair (1, 2) = (2, 1)

The point (3, 4) is different from the point (4, 3).

Let A = {1, 2, 3} and B = {5, 6}, what is the cartesian products of these two sets?

A × B = ?

Let A = {1, 2, 3, 4}, and B = {5, 6, 7}, Then the number of elements in the cartesian product set A x B = 

Two ordered pairs (a, b) and (c, d ) are equal if a = 1($) and b = 2($)

Let A and B be two nonempty sets.

A function ƒ from A to B is a subset of 1($)(involving the entire set 2($)) with the property that each “a” belonging to A, belongs to precisely one ordered pair (a, b), in the subset of A x B, under consideration. In other words, a function ƒ from set A to set B consists of a set of ordered pairs (a, b) A x B such that no two order pairs have the same 3($) element.

A function ƒ from set A to set B (written as ƒ: A → B) is a rule of correspondence that associates to each element of A, one and only one element of B.

1. Each element of 1($) need not be in the association, but every element of 2($) must be involved in it.
2. One element of 3($) cannot be associated with more than one element of 4($), but one element of 5($) may correspond to two or more elements of 6($).

Let ƒ be a function from set A to set B ( ƒ: A → B), then

The (entire) set A is called the 1($) of ƒ.

Let ƒ be a function from set A to set B ( ƒ: A → B), then

The (entire) set B is called the 1($) of ƒ.

Let ƒ be a function from set A to set B (ƒ: A → B), then

An element of B that corresponds to some element x of A is denoted by ƒ(x), and it is called the 1($) of x under ƒ.

Let ƒ be a function from set A to set B (ƒ: A → B), then

The set of all images constitutes the 1($) of ƒ.  The 2($) of ƒ is denoted by ƒ(A) and it is a 3($) of set B.  In other words ƒ(A) ⊆ B.