Most students will already know some of the material in this chapter, but even they should review some basics (sets, the standard number systems, and so on) so that we all use the same notation. We shall also introduce Venn diagrams, which will be a useful tool in more than one place.

Sets

All of mathematics rests on the foundations of set theory and numbers. We’ll start this chapter by reminding you of some basic definitions and notations and some further properties of numbers and sets.

A set is any collection of objects. Sets abound in our lives—most children have owned a train set (a collection of engines, cars, track pieces, and so on); you have sets of CDs, sets of colored pencils, and so on. You could talk about all your friends as a set, or all your clothes.

The objects in the collection are called the members or elements of the set. If x is a member of a set S, we write x ∈ S, and x ∉ S means that x is not a member of S.

One way of defining a set is to list all the elements, usually between braces; thus if S is the set consisting of the numbers 0, 1 and 3, we could write S = {0,1,3}.

Another method is to use the membership law of the set: for example, since the numbers 0, 1 and 3 are precisely the numbers that satisfy the equation x³ − 4x² + 3x = 0, we could write the set S as S = {x : x³ −4x² +3x = 0}

(which we read as “the set of all x such that x³ − 4x² + 3x = 0”). Often we use a vertical line instead of the colon in this expression, as in

                                        S = {x | x³ −4x² +3x = 0}.

This form is sometimes called set-builder notation.

Sample Problem 1.1 Write three different expressions for the set with elements 1 and −1.

Solution. Three possibilities are {1,−1},{x : x² = 1}, and “the set of square roots of 1”. There are others.

Your Turn. Write three different expressions for the set with the three elements 1, 2 and 3.

The definition of a set does not allow for ordering of its elements, or for repetition of its elements. For example, {1,2,3},{1,3,2} and {1,2,3,1} all represent the same set. To handle problems that involve ordering, we define a sequence to be an ordered set. Sequences are denoted by using parentheses (round brackets) instead of the braces that we use for sets; (1,3,2) is the sequence with first element 1, second element 3 and third element 2, and is different from (1,2,3). Sequences can containrepetitions, and (1,2,1,3) is quite different from (1,2,3).