Set Theory

A set is a group of objects, numbers, and so forth. {1,2,3} is a set consisting of the numbers 1,2, and 3. Verbally, “3 is an element of the set {1,2,3}.” To show this symbolically, use the symbol ∈, which is read as “is an element of” or “is a member of.” Therefore, you could have written:

3 ∈ {1,2,3}

Special sets

A subset is a set contained within another set, or it can be the entire set itself. The set {1,2} is a subset of the set {1,2,3}, and the set {1,2,3} is a subset of the set {1,2,3}. When the subset is missing some elements that are in the set it is being compared to, it is a proper subset. When the subset is the set itself, it is an improper subset. The symbol used to indicate “is a proper subset of” is ⊂. When there is the possibility of using an improper subset, the symbol used is ⊆. Therefore, {1,2} ⊂ {1,2,3} and {1,2,3} ⊆ {1,2,3}. The universal set is the general category set, or the set of all those elements under consideration. The empty set, or null set, is the set with no elements or members. The empty set, or null set, is represented by ⊘, or { }. However, it is never represented by {⊘}.

Both the universal set and the empty set are subsets of every set.

Describing sets

Rule is a method of naming a set by describing its elements.

{ x: x > 3, x is a whole number} describes the set with elements 4, 5, 6,…. Therefore, { x: x > 3, x is a whole number} is the same as {4,5,6,…}. { x: x > 3} describes all numbers greater than 3. This set of numbers cannot be represented as a list and is represented using a number line graph.

Roster is a method of naming a set by listing its members.

{1,2,3} is the set consisting of only the elements 1,2, and 3. There are many ways to represent this set using a rule. Two correct methods are as follows:

{ x: x < 4, x is a natural number}

{ x:0 < x < 4, x is a whole number}

An incorrect method would be { x:0 < x < 4} because this rule includes ALL numbers between 0 and 4, not just the numbers 1, 2, and 3.

Types of sets

Finite sets have a countable number of elements. For example, { a,b,c,d,e} is a set of five elements, thus it is a finite set. Infinite sets contain an uncountable number of elements. For example, {1,2,3, …} is a set with an infinite number of elements, thus it is an infinite set.

Comparing sets

Equal sets are those that have the exact same members — {1, 2, 3} = {3, 2, 1}. Equivalent sets are sets that have the same number of members — {1, 2, 3} | { a, b, c}.

Venn diagrams (and Euler circles) are ways of pictorially describing sets as shown in Figure 1.

Figure 1. A Venn diagram

figure

The A represents all the elements in the smaller oval; the B represents all the elements in the larger oval; and the C represents all the elements that are in both ovals at the same time.

Operations with sets

The union of two sets is a set containing all the numbers in those sets, but any duplicates are only written once. The symbol for finding the union of two sets is ∪.

Example 1

Find the union {1,2,3} ∪ {3,4,5}.

{1,2,3} ∪ {3,4,5} = {1,2,3,4,5}

The union of the set with members 1, 2, 3 together with the set with members 3, 4, 5 is the set with members 1, 2, 3, 4, 5.

The intersection of two sets is a set containing only the members that are in each set at the same time. The symbol for finding the intersection of two sets is ∩.

Example 2

Find the intersection {1,2,3} ∩ {3,4,5}.

{1,2,3} ∩ {3,4,5} = {3}

The intersection of the set with members 1, 2, 3 together with the set with members 3, 4, 5 is the set that has only the 3.

If you were to let the set with {1,2,3} be set A, and the set with {3,4,5} be set B, then you could use Venn diagrams to illustrate the situation (see Figure 2).

Figure 2. Intersection of set A and set B

figure

The union will be all the numbers represented in the diagram, {1,2,3,4,5}.The intersection would be where the two ovals overlap in the diagram, {3}.

Example 3

Find {1,2,3} ∩ {4,5}.

Since there are no members that are in both sets at the same time, then {1,2,3} ∩ {4,5} = ⊘.

The intersection of the set with members 1, 2, 3 together with the set with members 4, 5 is the empty set, or null set. There are no members in both sets at the same time.