10 CHAPTER 1. PRELIMINARIES

Definition 1.2.1 Consider two sets, D and R along with a rule which assigns a uniqueelement of R to every element of D. This rule is called a function and it is denoted by aletter such as f. Given x ∈ D, f (x) is the name of the thing in R which results from doingf to x. Then D is called the domain of f. In order to specify that D pertains to f , thenotation D (f) may be used. The set R is sometimes called the range of f. These days itis referred to as the codomain. The set of all elements of R which are of the form f (x)for some x ∈ D is therefore, a subset of R. This is sometimes referred to as the image off . When this set equals R, the function f is said to be onto, also surjective. If wheneverx ̸= y it follows f (x) ̸= f (y), the function is called one to one. , also injective It iscommon notation to write f : D 7→ R to denote the situation just described in this definitionwhere f is a function defined on a domain D which has values in a codomain R. Sometimes

you may also see something like Df7→ R to denote the same thing.

1.3 The Number Line and Algebra of the Real Num-bers

Next, consider the real numbers, denoted by R, as a line extending infinitely far in bothdirections. In this book, the notation, ≡ indicates something is being defined. Thus theintegers are defined as

Z ≡{· · · − 1, 0, 1, · · · } ,

the natural numbers,N ≡ {1, 2, · · · }

and the rational numbers, defined as the numbers which are the quotient of two integers.

Q ≡{mn

such that m,n ∈ Z, n ̸= 0}

are each subsets of R as indicated in the following picture.

0

1/2

1 2 3 4−1−2−3−4

As shown in the picture, 12 is half way between the number 0 and the number, 1. By

analogy, you can see where to place all the other rational numbers. It is assumed that R hasthe following algebra properties, listed here as a collection of assertions called axioms. Theseproperties will not be proved which is why they are called axioms rather than theorems. Ingeneral, axioms are statements which are regarded as true. Often these are things whichare “self evident” either from experience or from some sort of intuition but this does nothave to be the case.

Axiom 1.3.1 x+ y = y + x, (commutative law for addition)

Axiom 1.3.2 x+ 0 = x, (additive identity).

Axiom 1.3.3 For each x ∈ R, there exists −x ∈ R such that x + (−x) = 0, (existence ofadditive inverse).

Axiom 1.3.4 (x+ y) + z = x+ (y + z) , (associative law for addition).