Set Theory

Set theory is a fundamental concept throughout all of mathematics. This branch of mathematics forms a foundation for other topics. 

Intuitively a set is a collection of objects, which are called elements. Although this seems like a simple idea, it has some far-reaching consequences. 

Elements

The elements of a set can really be anything – numbers, states, cars, people or even other sets are all possibilities for elements. Just about anything that can be collected together may be used to form a set, though there are some things we need to be careful about.

Equal Sets

Elements of a set are either in a set or not in a set. We may describe a set by a defining property, or we may list the elements in the set. The order that they are listed is not important. So the sets {1, 2, 3} and {1, 3, 2} are equal sets, because they both contain the same elements.

Two Special Sets

Two sets deserve special mention. The first is the universal set, typically denoted U. This set is all of the elements that we may choose from. This set may be different from one setting to the next. For example, one universal set may be the set of real numbers whereas for another problem the universal set may be the whole numbers {0, 1, 2,...}. 

The other set that requires some attention is called the empty set. The empty set is the unique set is the set with no elements. We can write this as { } and denote this set by the symbol ∅.

Subsets and the Power Set

A collection of some of the elements of a set A is called a subset of A. We say that A is a subset of B if and only if every element of A is also an element of B. If there are a finite number n of elements in a set, then there are a total of 2ⁿ subsets of A. This collection of all of the subsets of A is a set that is called the power set of A.

Set Operations

Just as we can perform operations such as addition - on two numbers to obtain a new number, set theory operations are used to form a set from two other sets. There are a number of operations, but nearly all are composed from the following three operations:

• Union – A union signifies a bringing together. The union of the sets A and B consists of the elements that are in either A or B.
• Intersection - An intersection is where two things meet. The intersection of the sets A and B consists of the elements that in both A and B.
• Complement - The complement of the set A consists of all of the elements in the universal set that are not elements of A.

Venn Diagrams

One tool that is helpful in depicting the relationship between different sets is called a Venn diagram. A rectangle represents the universal set for our problem. Each set is represented with a circle. If the circles overlap with one another, then this illustrates the intersection of our two sets. 

Applications of Set Theory

Set theory is used throughout mathematics. It is used as a foundation for many subfields of mathematics. In the areas pertaining to statistics, it is particularly used in probability. Much of the concepts in probability are derived from the consequences of set theory. Indeed, one way to state the axioms of probability involves set theory.