Section 2.1 Set Concepts

What You Will Learn Upon completion of this section, you will be able to: Understand the use of description, roster form, and set-builder notation to indicate a set. Understand the fundamental set concepts including finite sets, infinite sets, equal sets, equivalent sets, the cardinal number of set, the null or empty set, and universal sets.

Set A set is a collection of objects, which are called elements or members of the set. Three methods used to indicate a set: Description Roster form Set-builder notation

Well-defined Set A set is well defined if its contents can be clearly determined. Example: The set of U.S. presidents is a well defined set because its contents, the presidents, can be named.

Example 1: Description of Sets Write a description of the set containing the elements January, February, March, April, May, June, July, August, September, October, November, December.

Example 1: Description of Sets Solution The set is the months of the year.

Roster Form Listing the elements of a set inside a pair of braces, { }, is called roster form. Example {1, 2, 3} is the notation for the set whose elements are 1, 2, and 3. (1, 2, 3) and [1, 2, 3] are not sets because parentheses and brackets do not indicate a set.

Naming of Sets Sets are generally named with capital letters. Definition: Natural Numbers The set of natural numbers or counting numbers is N. N = {1, 2, 3, 4, 5, …}

Example 2: Roster Form of Sets Express the following in roster form. a) Set A is the set of natural numbers less than 6. Solution: a) A = {1, 2, 3, 4, 5}.

Example 2: Roster Form of Sets Express the following in roster form. b) Set B is the set of natural numbers less than or equal to 45. Solution: b) B = {1, 2, 3, 4, …, 45}

Example 2: Roster Form of Sets Express the following in roster form. c) Set P is the set of planets in Earth’s solar system. Solution: c) P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}

Set Symbols The symbol ∈, read “is an element of,” is used to indicate membership in a set. The symbol ∉ means “is not an element of.”

Set-Builder Notation (or Set-Generator Notation) A formal statement that describes the members of a set is written between the braces. A variable may represent any one of the members of the set.

Example 4: Using Set-Builder Notation a) Write set B = {1, 2, 3, 4, 5} in set-builder notation. b) Write in words, how you would read set B in set-builder notation.

Example 4: Using Set-Builder Notation Solution a) or b) The set of all x such that x is a natural number and x is less than 6.

Example 6: Set-Builder Notation to Roster Form Write set in roster form. Solution A = {2, 3, 4, 5, 6, 7}

Finite Set A set that contains no elements or the number of elements in the set is a natural number. Example: Set B = {2, 4, 6, 8, 10} is a finite set because the number of elements in the set is 5, and 5 is a natural number.

Infinite Set A set that is not finite is said to be infinite. The set of counting numbers is an example of an infinite set.

Equal Sets Set A is equal to set B, symbolized by A = B, if and only if set A and set B contain exactly the same elements. Example: { 1, 2, 3 } = { 3, 1, 2 }

Cardinal Number The cardinal number of set A, symbolized n(A), is the number of elements in set A. Example: A = { 1, 2, 3 } and B = {England, Brazil, Japan} have cardinal number 3, n(A) = 3 and n(B) = 3

Equivalent Sets Set A is equivalent to set B if and only if n(A) = n(B). Example: D = { a, b, c }; E = {apple, orange, pear} n(D) = n(E) = 3 So set A is equivalent to set B.

Equivalent Sets - Equal Sets Any sets that are equal must also be equivalent. Not all sets that are equivalent are equal. Example: D = { a, b, c }; E = {apple, orange, pear} n(D) = n(E) = 3; so set A is equivalent to set B, but the sets are NOT equal

One-to-one Correspondence Set A and set B can be placed in one-to-one correspondence if every element of set A can be matched with exactly one element of set B and every element of set B can be matched with exactly one element of set A.

One-to-one Correspondence Consider set S states, and set C, state capitals. S = {North Carolina, Georgia, South Carolina, Florida} C = {Columbia, Raleigh, Tallahassee, Atlanta} Two different one-to-one correspondences for sets S and C are:

One-to-one Correspondence S = {No Carolina, Georgia, So Carolina, Florida} C = {Columbia, Raleigh, Tallahassee, Atlanta} S = {No Carolina, Georgia, So Carolina, Florida} C = {Columbia, Raleigh, Tallahassee, Atlanta}

One-to-one Correspondence Other one-to-one correspondences between sets S and C are possible. Do you know which capital goes with which state?

Null or Empty Set The set that contains no elements is called the empty set or null set and is symbolized by

Null or Empty Set Note that {∅} is not the empty set. This set contains the element ∅ and has a cardinality of 1. The set {0} is also not the empty set because it contains the element 0. It has a cardinality of 1.

Universal Set The universal set, symbolized by U, contains all of the elements for any specific discussion. When the universal set is given, only the elements in the universal set may be considered when working with the problem.

Universal Set Example If the universal set is defined as U = {1, 2, 3, 4, ,…,10}, then only the natural numbers 1 through 10 may be used in that problem.