Download - 2.1 Sets
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SetsSection 2.1
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Section SummaryDefinition of setsDescribing Sets
Roster MethodSet-Builder Notation
Some Important Sets in MathematicsEmpty Set and Universal SetSubsets and Set EqualityCardinality of SetsTuplesCartesian Product
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SetsA set is collection of objects.
the students in this class the chairs in this room
The contents have to be clearly determinedThe objects in a set are called the elements, or
members of the set. A set is said to contain its elements.
Set notation uses { }The notation a ∈ A denotes that a is an element
of the set A.If a is not a member of A, write a ∉ A
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Describing a Set: Roster MethodSets are named with capital lettersRoster method is listing all elementsS = {a,b,c,d}Order not important S = {a,b,c,d} = {b,c,a,d}Each distinct object is either a member or not;
listing more than once does not change the set. S = {a,b,c,d} = {a,b,c,b,c,d}Elipses (…) may be used to describe a set without listing all of the members when the pattern is clear. S = {a,b,c,d, …,z }
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Roster MethodSet of all vowels in the English alphabet: V = {a,e,i,o,u}Set of all odd positive integers less than 10: O = {1,3,5,7,9}Set of all positive integers less than 100: S = {1,2,3,…,99} Set of all integers less than 0: S = {…., -3,-2,-1}
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Tedious or ImpossibleRoster method is sometime tedious or
impossibleExample: (tedious)
L={x | x is a lowercase letter of the alphabet} L= {a, b, c … z}
Example: (impossible)Z = {x | x is a positive number}
Z = {1, 2, 3, 4 …}
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Some Important SetsN = natural numbers = {0,1,2,3….}Z = integers = {…,-3,-2,-1,0,1,2,3,…}Z⁺ = positive integers = {1,2,3,…..}R = set of real numbersR+ = set of positive real numbersC = set of complex numbers.Q = set of rational numbers
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Set-Builder NotationSpecify the property or properties that all
members must satisfy: S = {x | x is a positive integer less than 100}In word: The set S is all the elements x such that x is positive and less than 100 O = {x | x is an odd positive integer less than 10} O = {x ∈ Z⁺ | x is odd and x < 10}Positive rational numbers: Q+ = {x ∈ R | x = p/q, for some positive integers p,q}
Such thatAllElementsx
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Set-Builder Notation
Ex:Express Set A={x | x is a month begins with the letter M} using the roster method.
Word Description
Roster Method
Set-Builder Notation
W is the set of all days of the week
W= {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
W={x | x is a day of the week}
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Some things to rememberSets can be elements of sets. {{1,2,3},a, {b,c}} {N,Z,Q,R}The empty set is different from a set
containing the empty set. ∅ ≠ { ∅ } { }
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Example: Recognize the empty set:1. {0}
No
2. 0No, it is not a set
3. {x | x is a number less than 4 and greater than 10}
No
4. {x | x is a square with 3 sides}No
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Notations for Set Membersship∈
Is an element of ∈
Is not an element ofExample: Determine if each statement is true or false.
r ∈ {a, b, c … z} True
7 ∈ {1, 2, 3, 4, 5} False
{a} ∈ {a, b} False
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Set Equality Definition: Two sets are equal if and only if
they have the same elements. Therefore if A and B are sets, then A and B are
equal if and only if . We write A = B if A and B are equal sets.
{1,3,5} = {3, 5, 1} {1,5,5,5,3,3,1} = {1,3,5}
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Subsets Definition: The set A is a subset of B, if and
only if every element of A is also an element of B. The notation A ⊆ B is used to indicate that A is a subset of the set B. A ⊆ B holds if and only if is true.
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Showing a Set is or is not a Subset of Another SetShowing that A is a Subset of B: To show that A ⊆ B, show that if x belongs to A, then x also belongs to B.Showing that A is not a Subset of B: To
show that A is not a subset of B, A ⊈ B, find an element x ∈ A with x ∉ B. (Such an x is a counterexample to the claim that x ∈ A implies x ∈ B.) Examples: The set of all computer science majors at your school is a subset of all students at your school.
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Another look at Equality of SetsRecall that two sets A and B are equal,
denoted by A = B, iff
Using logical equivalences we have that A = B iff
This is equivalent to A ⊆ B and B ⊆ A
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Proper Subsets Definition: If A ⊆ B, but A ≠B, then we say A
is a proper subset of B, denoted by A ⊂ B. If A ⊂ B, then
is true.
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Set Cardinality Definition: If there are exactly n distinct elements in S where n is a
nonnegative integer, we say that S is finite. Otherwise it is infinite. Definition: The cardinality of a finite set A, denoted by |A|, is the
number of elements of A. Examples:1. |ø| = 02. Let S be the letters of the English alphabet
Then |S| = 263. C= {1,2,3}
|C|=34. D= {1, 1, 2, 3, 3} |D|= 35. E={ø}
|E|= 1The set of integers is infinite.
Finite: endsInfinite: no end
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Universal Set and Venn Diagram The universal set U: is the set containing
everything currently under consideration. Contents depend on the context.
Venn Diagram: visual relationship among setsUses circles and rectangles
U
Venn Diagram
a e i o u
V
John Venn (1834-1923)Cambridge, UK
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ExampleUse the Venn Diagram to determine each of the
following sets:
a) UU={$, 5, s, w}
b) BB={3}
UB
5
$
3s
w
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Representing 2 sets in a Venn Diagram 4 different ways
UA
Disjoint Set
Proper SubsetUAB
B
U
A=B
Equal Sets
Set with some common elements
UA B
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ExampleUse the Venn Diagram to determine each of the
following sets:a) U
U={a, b, c, d, e, f}b) B
B={c, d}
c) The set of elements in A but not B
{a, b}
d) The set of elements in U but not B{a, b, e, f}
e) The set of elements both in A and B{c}
UB
cb
a
f
e
A
d
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ExampleUse the Venn Diagram to illustrate the
relationship: A ⊆ B and B ⊆ C
U
BAC
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Power Sets Definition: The set of all subsets of a set A,
denoted P(A), is called the power set of A. Example: If A = {a,b} then P(A) = {ø, {a},{b},{a,b}}
If a set has n elements, then the cardinality of the power set is 2ⁿ.
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Cartesian Product Definition: The Cartesian Product of two sets A
and B, denoted by A × B is the set of ordered pairs (a,b) where a ∈ A and b ∈ B .
Example: A = {a,b} B = {1,2,3} A × B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)}
Definition: A subset R of the Cartesian product A × B is called a relation from the set A to the set B.
René Descartes (1596-1650)
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Cartesian Product
Example: What is A × B × C where A = {0,1}, B = {1,2} and C = {0,1,2}
Solution: A × B × C = {(0,1,0), (0,1,1), (0,1,2),(0,2,0), (0,2,1), (0,2,2),(1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,1,2)}
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Let A ={a, b, c}, B= {x, y}, and C={0, 1}Find:a) A × B × C
{(a, x, 0), (a, x, 1), (a, y, 0), (a, y, 1), (b, x, 0), (b, x, 1), (b, y, 0), (b, y, 1), (c, x, 0), (c, x, 1), (c, y, 0), (c, y, 1)}
b) C × B × A{(0, x, a), (0, x, b), (0, x, c), (0, y, a), (0, y, b), (0, y, c), (1, x, a), (1, x, b), (1, x, c), (1, y, a), (1, y, b), (1, y, c)}