Sets

A set is a collection of items. Mathematicians usually refer to the items as elements. A set is said to contain elements. The order of the elements makes no difference. A set emerges from the elements that it contains. When you define a set, such as the set of integers or counting numbers, you define the set of all elements it can contain. You might not name all the elements, however, because a set can consist of a finite or infinite number of elements. You still define the condition by which you can determine whether a given item is or is not an element of the set.

Elements

For example, you might picture a set of prime numbers. A prime number is a positive integer that you can generate by multiplication using only the number itself and 1. The other number must be distinct from the prime number. For this reason, 1 is not a prime number.

Using such a definition, you can describe an infinite set of numbers. At the same time, you can create sets of prime numbers using definitions that are more restricted. Here is the set of prime numbers less than or equal to 47:

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}

You identify a set using opening and closing curly braces, and you separate the elements in a set using commas. To denote that a number is a member of a set, you provide your set with a name. Although no strict rule applies, mathematicians commonly represent the names of sets with italicized capital letters. You might see the following, for example:

A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}

You can employ a small italic letter to represent the element symbolically. You use to designate that an element “is a member of” a set. If a equals 3, then

a A and 3 A

The number 1 is not a prime number, nor is 4. Assume that the value of a equals 1. To designate that 1 and 4 are not members of set A, you use the following notation:

a A 4 A

If it so happens that you have a set that has no elements, then the set is said to be empty. To designate an empty set, you can use the symbol Ø: A = Ø.

Subsets and Supersets

Any collection of items can be a set. At the same time, if you have a set, you can also create a subset. One set is a subset of another if the elements it contains are also contained by the superset. Consider the set of prime numbers less than or equal to 17. Designate that as set B:

B = {2, 3, 5, 7, 11, 13, 17}

Set B is a subset of set A (defined in the previous section as prime numbers less than or equal to 47), because each element that is in set B is also in set A. The notation you employ to denote that set B is a subset of set A is as follows:

You can also designate that A is a superset of B:

Expressions to Define Sets

When you define a set, you can create a list of the elements of a set, separate them using commas, and enclose the result in opening and closing curly braces. Such an approach to set creation suffices in many practical situations. For other situations, you can use a form of notation that allows you to describe the conditions of membership. In such situations, you make use of a vertical bar and a few other symbols.

Consider a situation in which you want to designate any number that is an element of set A. To accomplish this task, you use the following expression:

{a | a A}

The previous expression reads, “a such that a is an element of set A.” If you want to designate numbers less than 17, you can write:

{a | a < 17}

If you want to designate that a number a is less than 17 and is also an element of set A, then you can use a logical “and” symbol (∧):

{a | a < 17 ∧ a | a A}

This expression reads, “a such that a is less than 17 and a such that a is an element of set A.”

Unions and Intersections

Consider sets A and C:

A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}

C = {1, 2, 3, 5, 7, 10, 11, 12, 14, 17}

You can find some of the elements of set C in set A and some of the elements of set A in set C. In other words, between sets A and C exists a group of elements that are common to both sets. When you have a set of elements that are common to both sets, you find the intersection of the two sets. To indicate the intersection of two sets, you employ the ∩ symbol:

A ∩ C

More explicitly, you indicate the members of the set using an equation:

A ∩ C = {2, 3, 5, 7, 11, 17}

Not all of the elements in set C are in set A, and not all of the elements in set A are in set C. When you combine the elements of the two sets so that you have the elements of both sets without duplicates, you create the union of the two sets. To indicate the union of two sets, you employ the ∪ symbol:

A ∪ C

To show the union explicitly, you use the same approach you use when showing an intersection explicitly:

A ∪ C = {1, 2, 3, 5, 7, 10, 11, 12, 13, 14, 17, 19, 23, 29, 31, 37, 41, 43, 47}

Disjunctions

Unions and intersections allow you to explore how different sets share elements. Situations also arise in which you find sets that share no common elements, but you still want to show that they constitute a set. The logical term that you apply to such situations is or. Consider, for example, the set of all elements contained in sets D and E:

D = {1, 2, 3, 4}

E = {6, 7, 8, 9}

Set D contains numbers that are less than 5 and greater than 0, whereas set E contains numbers that are greater than 5 and less than 10. If you want to create a set in which you can account for both of these sets, you can start with a logical expression that employs the or symbol ( ). For example, if a expresses any number of the two sets, then

{a | a < 5 ∧ a | a > 0} {a | a > 5 ∧ a | a < 10}

This expression allows you to say that the set includes numbers less than 5 and greater than 0 or numbers greater than 5 and less than 10. It so happens, however, you can create a set F that consists of a union of these two sets:

F = {a | a < 5 ∧ a | a > 0} ∪ {a | a > 5 ∧ a | a < 10}