Function

A function f from a set X to a set Y is a relationship between elements of X and elements of Y with the property that each element of X is related to a unique element of Y. The notation f : X ® Y means that f is a function from X to Y. X is called the domain of f and Y is called the co-domain of f.
Given an element x in X, there is a unique element y in Y that is related to x. We can think of x as input and y as the related output. We then say "f sends x to y" and write x    ^f® Y. The unique element y to which f sends x is denoted
                                          f(x) and is called f of x,
                                                                   the value of f at x,or
                                                                   the image of x under f.
The set of all values of f taken together is called the range of f or the image of X under f. Symbolically:
                                          range of f = {y Î Y | y = f(x), for some x in X}.
Given an element y in Y, there may exist elements in X with y as their image. The set of all such elements is called the inverse image of y. Symbolically:
                                          inverse image of y = {xÎ X | f(x) = y}.

 

Equals Function

Suppose f and g are functions from X to Y. Then f equals g, written f = g, if, and only if,
                                         f(x) = g(x) for all xÎ X.

 

Greog Cantor "In 1873 Cantor proved the rational numbers countable, i.e. they may be placed in one-one correspondence with the natural numbers. He also showed that the algebraic numbers, i.e. the numbers which are roots of polynomial equations with integer coefficients, were countable. However his attempts to decide whether the real numbers were countable proved harder. He had proved that the real numbers were not countable by December 1873 and published this in a paper in 1874. It is in this paper that the idea of a one-one correspondence appears for the first time, but it is only implicit in this work."

Countably Infinite 

A set is called countably infinite iff it has the same cardinality as the set of positive integers Z⁺ . A set is called countable iff it is finite or countably infinite. A set that is not countable is called uncountable.

Ex. 7.6.1 Countability of Z, the Set of All Integers (p 414)
The set Z of all integers is certainly not finite. So if it is countable, it must be because it is countably infinite. To show Z is countably infinite, find a function that maps from the set of positive integers Z⁺ to Z that is one-to-one and onto.

Ex. 7.6.2 Countability of 2 Z, the Set of All Even Integers (p 415)
Consider the function h from Z to 2Z defined as follows:

h(n) = 2n for all n in Z.

A partial arrow diagram for h reveals clearly exhibits this relationship. The function h is certainly one-to-one and onto (see exercises 8b and 34 of 7.3). Now from 7.6.1 we know Z⁺ has the same cardinality as Z. So by the transitive property of cardinality, Z⁺ has the same cardinality as 2Z. It follows by the definition of countably infinite that 2Z is countably infinite and thus countable.

E. 7.6.3 The Set of All Positive Rational Numbers Is Countable