2.4. ORDER 13

{1,2,3,8} ⊆ {1,2,3,4,5,8} . The same statement about the two sets may also be writtenas {1,2,3,4,5,8} ⊇ {1,2,3,8}.

The union of two sets is the set consisting of everything which is contained in at leastone of the sets, A or B. As an example of the union of two sets, {1,2,3,8}∪{3,4,7,8} ={1,2,3,4,7,8} because these numbers are those which are in at least one of the two sets.In general

A∪B≡ {x : x ∈ A or x ∈ B} .Be sure you understand that something which is in both A and B is in the union. It is not anexclusive or.

The intersection of two sets, A and B consists of everything which is in both of the sets.Thus {1,2,3,8}∩{3,4,7,8}= {3,8} because 3 and 8 are those elements the two sets havein common. In general,

A∩B≡ {x : x ∈ A and x ∈ B} .When with real numbers, [a,b] denotes the set of real numbers x, such that a ≤ x ≤ b

and [a,b) denotes the set of real numbers such that a ≤ x < b. (a,b) consists of the setof real numbers, x such that a < x < b and (a,b] indicates the set of numbers, x such thata < x≤ b. [a,∞) means the set of all numbers, x such that x≥ a and (−∞,a] means the setof all real numbers which are less than or equal to a. These sorts of sets of real numbersare called intervals. The two points, a and b are called endpoints of the interval. Otherintervals such as (−∞,b) are defined by analogy to what was just explained. In general, thecurved parenthesis indicates the end point it sits next to is not included while the squareparenthesis indicates this end point is included. The reason that there will always be acurved parenthesis next to ∞ or −∞ is that these are not real numbers. Therefore, theycannot be included in any set of real numbers. It is assumed that the reader is alreadyfamiliar with order which is discussed in the next section more carefully. The emphasishere is on the geometric significance of these intervals. That is [a,b) consists of all pointsof the number line which are to the right of a possibly equaling a and to the left of b. In theabove description, I have used the usual description of this set in terms of order.

A special set which needs to be given a name is the empty set also called the null set,denoted by /0. Thus /0 is defined as the set which has no elements in it. Mathematicianslike to say the empty set is a subset of every set. The reason they say this is that if it werenot so, there would have to exist a set A, such that /0 has something in it which is not in A.However, /0 has nothing in it and so the least intellectual discomfort is achieved by saying/0⊆ A.

If A and B are two sets, A\B denotes the set of things which are in A but not in B. Thus

A\B≡ {x ∈ A : x /∈ B} .

Set notation is used whenever convenient.

2.4 OrderThe real numbers also have an order defined on them. This order may be defined by ref-erence to the positive real numbers, those to the right of 0 on the number line, denoted byR+ which is assumed to satisfy the following axioms.

Axiom 2.4.1 The sum of two positive real numbers is positive.

Axiom 2.4.2 The product of two positive real numbers is positive.