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Section 1.5. Venn Diagrams and Set Operations. 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.

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section 1 5

Section 1.5

Venn Diagrams and Set Operations

objectives
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 andandor.
  • Use the formula for n(A B).
key terms
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.
key terms continued
Key Terms (Continued)
  • Union of Sets:the union of sets A and B, symbolized by AB, 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.
sets take on different forms
Sets Take on Different Forms

Disjoint

B

A

A

B

Proper

Equal

Overlapping

A

A

=

B

Sets with some common elements.

B

overlapping sets
Overlapping Sets

Four Regions

A

B

Region I: elements in set A only.Region II: elements in set A and set B

Region III: elements in set B only.Region IV: elements that do not belong in set A or set B.

slide7
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.
example 1
Example 1:
  • Describe the Universal set that includes all elements in the given sets.
  • Set A= {Wm. Shakespeare, Charles Dickens}Set B = {Mark Twain, Robert Louis Stevenson}
  • Set A = {Pepsi, Sprite, Dr. Pepper}Set B = {Coca Cola, Seven-Up}
example 2
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}
  • Find B’
  • Find C’
example 3 a b
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}
  • Find A
  • Find B
  • Find ∩
example 4 b c
Example 4: BC
  • U = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 5, 7}B = {1, 2, 3}C = {2, 3, 4, 5, 6}
  • Find B
  • Find C
  • Find
example 5 b c
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}
  • Find B’ C
  • Find ∩
section 1 5 assignments
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
example 6 a b
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}
  • Find A’
  • Find B’
  • Find A’ B’
example 7 a b c
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}
  • Find B
  • Find C
  • Find ∩
  • Find A’
  • Find
example 8
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
example 9
Example 9:
  • Using the same information from Example 8.
  • (EH) ∩ B
key terms1
Key Terms
  • Difference of Sets: the set of elements that are in B but not in A. This is denoted by B – A.
example 10 using set difference
Example 10: Using Set Difference
  • Find {3, 6, 9, 12, 15} – {x/x is an odd integer}
  • M = {jo, st}, W = {ba, be, ca, st}…Find M – W
section 1 5 assignments1
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.
section 1 5 con t
Section 1.5 con’t

Venn Diagrams and Set Operations with Three Sets

demorgan s law
DeMorgan’s Law
  • (A B)’ = A’ ∩ B’
  • (A ∩ B)’ = A’ B’
example 11
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}
example 12
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}
example 13
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}
example 14
Example 14:
  • Which regions represent set C?
example 15
Example 15:
  • Which regions represent B C?
example 16
Example 16:
  • Use the Venn diagram to represent each set in roster form.

4 5

6

7 8

9

example 17
Example 17:
  • Use the Venn diagram to represent the set in roster form.

4 5

6

7 8

9

example 18
Example 18:
  • Construct a Venn diagram using the following information.
example 19
Example 19:
  • Determine if the sets are equal using a Venn diagram.
example 20
Example 20:
  • Determine if the sets are equal using a Venn diagram.
section 1 5 assignments2
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.