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Perform set operations: union, intersection, complement, and cross product.
Use Venn diagrams.
SET THEORY
1
Roster form, set-builder notation, empty set, universal set, complement of a set, intersection, Union.
SET THEORY
A set is a collection of objects. These objects are called elements or member of the set. A set should be well defined ( no ambiguity).
},,,,{ uoieaC
Capital Letters such as A, B, C..
Braces
Elements
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Set-builder notation: It describes the properties an element must have to be included an a set.
Example 1:
number natural a is | xxN
,...}5,4,3,2,1{N
Set-builder notation:
The set of all numbers x such that x is a natural number.
Roster form:
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Roster form: list the elements of the set within braces.
Note: There is two ways to write a set:
Your turn: Write each set in roster form and in set-builder notation.
•M is the set of integers that are greater than 4.
4 -integeran is | xxM
...}3 ,2 ,1 ,0 ,1,2,3{ M
Roster form:
Set-builder notation:
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A subset is a set whose elements are also contained in another set. The symbol means “ is a subset of “ .
A B A is a subset of B
The empty set is a subset of any set. A
A
B
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Example 1: Let A={1,2,3,4,5…} and B={1,3,5,7…} Is B a subset of A?
Yes, B A Answer
Empty set: There are no elements in the set.
Note:
Intersection: If we have two sets A and B, then the intersection of A and B is a set of elements that are common to both A and B. The notation for intersection is
Example: Given the sets
}5,4,3,2,1{A
}8,6,4,2{B }4,2{
BA
BA
3.8 Intersections and unions of sets.
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Your turn: Given the sets:
37,34,27,14,7 57,47,37,27,17 NandM
37,27NM Answer
B 6
8
2 4
A
1 3
5
Union: If we have two sets A and B, then the union of A and B is a set of elements that are members of A ,or members of B, or members of both A and B. The notation for union is
Example: Given the sets
}5,4,3,2,1{A }8,6,4,2{B
}8,6,5,4,3,2,1{
BA
BA
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Your turn: Given the sets
14,12,10,8,7,6,4,2S 10,9,8,7 andT
14,12,10,9,8,7,6,4,2ST Answer
1
3 5 2 4
6
8
A B
Universal set: It is a set which contains all the elements being considered in the given problem. Notation: U
},,,,,{ fedcbaU
},,{ bcaA
},,{ abeB
},{ ba},,,{ ecba
BA
BA
UBA
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Given the sets. Complete the Venn Diagram.
Your turn
Answer
b
a
c e
d f
And now include all elements of Universal
Note: The complement of a set A is the set of all elements in the given universal set ,that are not in set A. Notation: AU
},,,,,{ fedcbaU Example: Given sets. Find A’
},,{ fedA
},,{ bcaA },,{ abeB
},,{ fdcB
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Your turn. Find B’.
Answer
Your turn: At Vanessa’s high school, 32 girls play volleyball and 35 girls play basketball. Of these, 11 play both volleyball and basketball.Which of these diagrams best represents these numbers?
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Your turn again
The Venn diagram below shows the number of students, out of a class of 30, who earned an A in mathematics (M) and an A in English (E)
EM
47 6
13
How many students:
1. Earned an A in mathematics?
2. Earned an A in English? 3. Earned an A in both mathematics and English?
4. Did NOT earn an A in math and did NOT earn an A in English? 5. Earned an A in math but NOT English? 6. Earned an A in English but NOT in math?
Answers: 1.11 3. 4 5. 7 2.10 4. 13 6. 6
11
Your turn #3
This Venn Diagram shows salad topping used for 100 salads in a cafeteria. B=bacon, CH=cheese, CR=croutons.
CHB
11
18
How many salads had:
1. Just bacon as a topping?
2. Just bacon and cheese?
3. Just croutons and bacon?
4. all three topping?
5. No topping?
9
107 8
25
12
CR Answers: 1. 11, 2. 9, 3. 7, 4. 10, 5. 18
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Example: If A = {1, 2, 3} and B = {a, b} find
Using the number 1 from set A:
The Cartesian product (read “A cross B”) of two sets A and B is defined as the set of all order pairs (a, b) where a is a member of A and b is a member of B.
BA
BA
BA = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b) }
(2, a), (2, b)
(3, a), (3, b)
(1, a), (1, b) Using the number 2 from set: A:
Using the number 3 from set A:
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CLASSWORK: Form K HOMEWORK: Form G
