Why Read This Book? - Index of

2 Chapter 0 Notation and Assumptions
How can you communicate to someone what the elements of a set are? There
are several ways.
1. List them. If there are only a few elements in the set, you can easily list them
all. Otherwise, you might start listing the elements and hope that the reader
can take the hint and figure out the pattern. For example,
(a) {1, 8,π,Monday}
(b) {0, 1, 2,...,40}
(c) {...,−6, −4, −2, 0, 2, 4, 6,...}
2. Provide a description of the criteria used to define whether an entity is to be
included. It works like this:
(a) {x : x is a real number and x>−1}
This notation should be read “the set of all x such that x is a real number
and x is greater than −1.” The indeterminate x is just a symbol chosen to
represent an arbitrary element of the set, so that any characteristics it must
have can be stated in terms of that symbol.
(b) {p/q : p and q are integers and q �= 0}
This is the set of all fractions, integer over integer, where it is expressly
stated that the denominator cannot be zero.
(c) {x : P(x)}
This is a generic form for this way of describing a set. The expression
P(x) represents some specified property that x must have in order to be in
the set.
Some of the sets we will use most are the following:
Empty set: ∅={} (the set with no elements)
Natural numbers: N ={1, 2, 3,...}
Whole numbers: W ={0, 1, 2, 3,...}
Integers: Z ={...,−3, −2, −1, 0, 1, 2, 3,...}
Rational numbers: Q ={p/q : p, q ∈ Z,q�= 0}
Real numbers: R (Explained in the next section)
Nonzero real numbers: R ×
Given two sets A and B, it just might happen that all elements of A are also
elements of B. We write this as A ⊆ B and say that A is a subset of B. Equivalently,
we may write B ⊇ A, and say that B is a superset of A. IfA is a subset of B, but
there are elements of B that are not in A, we say that A is a proper subset of B,
and write this A ⊂ B.