Chapter 3

Set Theory3.1 Basic Definitions

This chapter has more on set theory. Recall a set is a collection of things called elementsof the set. For example, the set of integers, the collection of signed whole numbers suchas 1,2,−4, etc. This set whose existence will be assumed is denoted by Z. Other setscould be the set of people in a family or the set of donuts in a display case at the store.Sometimes parentheses, { } specify a set by listing the things which are in the set betweenthe parentheses. For example the set of integers between−1 and 2, including these numberscould be denoted as {−1,0,1,2}. The notation signifying x is an element of a set S, iswritten as x ∈ S. Thus, 1 ∈ {−1,0,1,2,3}. Here are some axioms about sets.

Axiom 3.1.1 Two sets are equal if and only if they have the same elements.

Axiom 3.1.2 To every set A, and to every condition S (x) there corresponds a set B, whoseelements are exactly those elements x of A for which S (x) holds.

Axiom 3.1.3 For every collection of sets there exists a set that contains all the elementsthat belong to at least one set of the given collection.

Axiom 3.1.4 The Cartesian product of a nonempty family of nonempty sets is nonempty.

Axiom 3.1.5 If A is a set there exists a set P (A) , such that P (A) is the set of all subsetsof A. This is called the power set.

These axioms are referred to as the axiom of extension, axiom of specification, axiomof unions, axiom of choice, and axiom of powers respectively.

It seems fairly clear you should want to believe in the axiom of extension. It is merelysaying, for example, that {1,2,3} = {2,3,1} since these two sets have the same elementsin them. Similarly, it would seem you should be able to specify a new set from a given setusing some “condition” which can be used as a test to determine whether the element inquestion is in the set. For example, the set of all integers which are multiples of 2. This setcould be specified as follows.

{x ∈ Z : x = 2y for some y ∈ Z} .

In this notation, the colon is read as “such that” and in this case the condition is being amultiple of 2.

Another example of political interest, could be the set of all judges who are not judicialactivists. I think you can see this last is not a very precise condition since there is no wayto determine to everyone’s satisfaction whether a given judge is an activist. Also, justbecause something is grammatically correct does not mean it makes any sense. Forexample consider the following nonsense.

S = {x ∈ set of dogs : it is colder in the mountains than in the winter} .

So what is a condition?We will leave these sorts of considerations and assume our conditions “make sense”.

The axiom of unions states that for any collection of sets, there is a set consisting of all

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