section 2.1 set concepts
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Section 2.1 Set Concepts. What You Will Learn. Equality of sets Application of sets Infinite sets. Set. A set is a collection of objects, which are called elements or members of the set. Three methods of indicating a set: Description Roster form Set-builder notation. Well-defined Set. - PowerPoint PPT PresentationTRANSCRIPT
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 2.1
Set Concepts
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What You Will Learn
Equality of sets
Application of sets
Infinite sets
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Set
• A set is a collection of objects, which are called elements or members of the set.
• Three methods of indicating a set:• Description• Roster form• Set-builder notation
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Well-defined Set
A set is well defined if its contents can be clearly defined.
Example:The set of U.S. presidents is a well defined set. Its contents, the presidents, can be named.
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Example 1: Description of SetsWrite a description of the set containing the elements Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday.
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Example 1: Description of SetsSolution
The set is the days of the week.
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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.
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Naming of Sets
Sets are generally named with capital letters.
Definition: Natural NumbersThe set of natural numbers or counting numbers is N.
N = {1, 2, 3, 4, 5, …}
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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}
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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 80.
Solution:b) B = {1, 2, 3, 4, …, 80}
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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}
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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.”
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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.
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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.
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Example 4: Using Set-Builder NotationSolution
a) or
b) The set of all x such that x is a natural number and x is less than 6.
B x x N and x 6 B x x N and x 5
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Example 6: Set-Builder Notation to Roster FormWrite setin roster form.
SolutionA = {2, 3, 4, 5, 6, 7}
A x x N and 2 x 8
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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.
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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.
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Set A is equal to set B, symbolized by A = B, if and only if set A and set B contain exactly the same members.
Example: { 1, 2, 3 } = { 3, 1, 2 }
Equal Sets
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Cardinal NumberThe cardinal number of set A, symbolized n(A), is the number of elements in set A.Example:A = { 1, 2, 3 } andB = {England, Brazil, Japan}have cardinal number 3,n(A) = 3 and n(B) = 3
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Equivalent SetsSet 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) = 3So set A is equivalent to set B.
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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
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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.
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One-to-one CorrespondenceConsider 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:
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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}2.1-25
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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?
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Null or Empty Set
The set that contains no elements is called the empty set or null set and is symbolized by
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or .
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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.
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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.
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Universal Set
Example
If the universal set is defined asU = {1, 2, 3, 4, ,…,10}, then only the natural numbers 1 through 10 may be used in that problem.
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