s ection 1.5 venn diagrams and set operations. o bjectives understand the meaning of a universal set...
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SECTION 1.5Venn Diagrams and Set Operations
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OBJECTIVES Understand the meaning of a universal set and the basic
ideas of a Venn diagram.
Use Venn diagrams to visualize relationships between two sets.
Find the complement of a set.
Find the intersection and union of two sets.
Perform operations with sets.
Determine sets involving set operations from a Venn diagram.
Understand the meaning of and and or.
Use the formula for n(A B).
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KEY TERMS Universal Set: a set that contains all the
elements for any specific discussion, symbolized by .
Venn Diagrams: (named for British logician, John Venn) a rectangle is drawn to represent the .
Complement of a Set: complement of Set A is the set of all elements in the set that are not in set A; symbolized by A’.
Intersection of Sets: intersection of set A and B is the set of elements that A and B have in common, symbolized by A∩B.
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KEY TERMS (CONTINUED) Union of Sets: the union of sets A and B,
symbolized by A B, is the set of elements that are members of either A or B (or both).
And and or: the word “or” generally means union. The word “and” generally means intersection.
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SETS TAKE ON DIFFERENT FORMS
Disjoint
Proper
OverlappingEqual
A
A
A
A
B
B
B
B=Sets with
some common elements.
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OVERLAPPING SETS
RegionI
Region III
A B
Four Regions
Region I: elements in set A only.Region II: elements in set A and set BRegion III: elements in set B only.Region IV: elements that do not belong in set A or set B.
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NOTE: For any set A:
A ∩ =
A = A
Performing Set Operations: always begin by performing set operations inside parentheses; or just identify the elements in each set.
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EXAMPLE 1: Describe the Universal set that includes all
elements in the given sets.
a. Set A= {Wm. Shakespeare, Charles Dickens}Set B = {Mark Twain, Robert Louis Stevenson}
b. Set A = {Pepsi, Sprite, Dr. Pepper}Set B = {Coca Cola, Seven-Up}
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EXAMPLE 2: U = {a, b, c, d, e, f, g}, A = {a, b, f, g}, B =
{c, d, e}, C = {a, g}, and D = {a, b, c, d, e, f}
a. Find B’
b. Find C’
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EXAMPLE 3: A ∩ B U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}
1. Find A
2. Find B
3. Find ∩
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EXAMPLE 4: B C U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}
1. Find B
2. Find C
3. Find
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EXAMPLE 5: B’ ∩ C U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}
1. Find B’ C
2. Find ∩
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SECTION 1.5 ASSIGNMENTS Classwork:
TB pg. 46/1 - 10 Must write problems and show ALL work to receive
credit for this assignment.
Homework: Create Engrade Account
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EXAMPLE 6: A’ B’ U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}
1. Find A’
2. Find B’
3. Find A’ B’
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EXAMPLE 7: A’ (B ∩ C) U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}
1. Find B
2. Find C
3. Find ∩
4. Find A’
5. Find
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EXAMPLE 8: In order to increase its readership, a computer
magazine conducted a survey of people who have recently purchased a new computer and identified the following groups:
E = {x/x will use the computer for education}, B = {x/x will use the computer for business}, H = {y/y will use the computer for home management}
Use this information to describe verbally the following set.
E ∩ H
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EXAMPLE 9:
Using the same information from Example 8.
(E H) ∩ B
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KEY TERMS Difference of Sets: the set of elements
that are in B but not in A. This is denoted by B – A.
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EXAMPLE 10: USING SET DIFFERENCE
a. Find {3, 6, 9, 12, 15} – {x/x is an odd integer}
b. M = {jo, st}, W = {ba, be, ca, st}…Find M – W
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SECTION 1.5 ASSIGNMENTS Classwork:
TB pg. 46/13 – 24 all Must write problems and show ALL work to receive
credit for this assignment.
Homework: Do not forget to create Engrade account.
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SECTION 1.5 CON’TVenn Diagrams and Set Operations with Three Sets
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THREE SETS – 8 REGIONS
IIII
VII
II
IVV
VI
VIII
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DeMorgan’s Law (A B)’ = A’ ∩ B’
(A ∩ B)’ = A’ B’
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EXAMPLE 11: U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}
)( CBA
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EXAMPLE 12: U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}
)()( CABA
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EXAMPLE 13: U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}
)''()'( BCAC
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EXAMPLE 14: Which regions represent set C?
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EXAMPLE 15: Which regions represent B C?
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EXAMPLE 16: Use the Venn diagram to represent each set
in roster form.
)'( CB
1 2 3 10
11
12
4 5
7 8 9
6
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EXAMPLE 17: Use the Venn diagram to represent the set in
roster form.
1 2 3 10
11
12
4 5
7 8 9
6
)'( CBA
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EXAMPLE 18: Construct a Venn diagram using the following
information.
},,,,,,,,{
},,{
},,,,,{
},,,{
ihgfedcbaU
gfeC
ihfecbB
iheaA
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EXAMPLE 19: Determine if the sets are equal using a Venn
diagram.
BA' 'BA
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EXAMPLE 20: Determine if the sets are equal using a Venn
diagram.
)''( BA )'( BA
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SECTION 1.5 ASSIGNMENTS Classwork:
TB pg. 47/26 – 44 Even, and 57 – 64 All Must write problems and show ALL work to receive
credit for this assignment.
Homework: Do not forget to create Engrade account.