GMAT Exam: A Simple Guide To Arithmetic - Sets

Terminology
A set is a collection of objects. Sets are usually named with uppercase letters, such as A and B. The set’s objects are its elements or members.

In the roster notation for sets, the set’s elements, separated by commas, are listed between curly braces.

For instance, the set P, consisting of the first ten prime numbers, is P = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}. You write 11 ∈ P to mean 11 is an element of P.

The set that contains no elements is the empty set, denoted ∅.

The number of elements in set A is its cardinality, denoted |A| (also, #A). For instance, if A = {2, 3, 5}, |A| = 3.

Set Relationships
Two sets are equal if and only if they contain exactly the same elements, without regard to the order in which the elements are listed in the two sets or whether elements are repeated.

For instance, {2, 3, 5} = {2, 5, 3} = {2, 2, 3, 3, 5, 5}.

Set A is a subset of set B, written A ⊆ B, if every element of A is an element of B.

For example, {2, 5} ⊆ {2, 3, 5}.

In a discussion, the universal set (often denoted U) contains all the sets under consideration as subsets.

The union of two sets A and B, denoted A ∪ B, is the set of all elements that are in A or in B or in both.

Tip: The word or in this definition is inclusive; that is, or means “one or the other, or possibly both at the same time.”

For example, if A = {2, 3, 5} and B = {3, 5, 7, 11}, then A ∪ B = {2, 3, 5, 7, 11}.

Tip: When you form the union of two sets, do not list an element more than once, because it is unnecessary to do so.

The intersection of two sets A and B, denoted A ∩ B, is the set of all elements that are common to both sets.

For instance, if A = {2, 3, 5} and B = {3, 5, 7, 11}, then A ∩ B = {3, 5}.

Disjoint sets have no elements in common.

Their intersection is the empty set.

For instance, if A = {2, 3, 5} and C = {7, 11, 13, 17}, then A ∩ C = ∅ and A and C are disjoint.

The complement of set A, denoted AC (also, or ~A), is the set of all elements in the universal set U that are not in A.

For instance, if U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {2, 3, 5}, then AC = {1, 4, 6, 7, 8, 9}.

Venn diagrams visually depict set relationships.

In a Venn diagram, circles represent sets—with the exception that the universal set is represented by a rectangular region, enclosing everything else in the diagram. Shading depicts the results of relationships.

Here are examples.