Sets

We need some basic information about sets in order to study the logic and the axiomatic method. This is not a formal study of sets, but consists only of basic definitions and notation.

Braces and are used to name or enumerate sets. The roster method for naming sets is simply to list all of the elements of a set between a pair of braces. For example the set of integers 1, 2, 3, and 4 could be named

This does not work well for sets containing a large number of elements, though it can be used. The more common method for this is known as the set builder notation. A property is specified which is held by all objects in a set. P(x), read P of x, will denote a sentence referring to the variable x. For example,

• x=23
• x is an odd integer.
• .

The set of all objects x such that x satisfies P(x) is denoted by

The set can be named

From hence forth, the words object, element, and member mean the same thing when referring to sets. Sets will be denoted mainly by capital Roman letters and elements of the sets by small letters. The following have the same meaning:

• 
• a is in set A
• a is a member of set A
• a is an element of set A

Likewise, means that a is not an element of set A.

A is a subset of B if every element of A is also an element of B. The following have the same meaning:

• 
• Every element of A is an element of B
• If , then
• A is included in B
• B contains A
• A is a subset of B

Note that a set is always a subset of itself.

If A and B are sets, then we say that A=B if A and B represent the same set:

• A=B
• A and B are the same set
• A and B have the same members
• and

The set which contains no elements is known as the empty set, and is denoted by . Note that for each set A, .

The intersection of two sets A and B is the set of all elements common to both sets. The intersection is symbolized by or . The union of two sets A and B is the set of elements which are in A or B or both. The union is symbolized by or .