Section 1.5

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# Section 1.5 - PowerPoint PPT Presentation

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

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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.
• 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
• 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)
• 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

Disjoint

B

A

A

B

Proper

Equal

Overlapping

A

A

=

B

Sets with some common elements.

B

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.

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:
• 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:
• 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
• 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: 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
• 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
• Classwork:
• TB pg. 46/1 - 10
• Must write problems and show ALL work to receive credit for this assignment.
• Homework:
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)
• 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:
• 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:
• Using the same information from Example 8.
• (EH) ∩ B
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
• 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 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

Venn Diagrams and Set Operations with Three Sets

DeMorgan’s Law
• (A B)’ = A’ ∩ B’
• (A ∩ B)’ = A’ B’
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:
• 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:
• 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:
• Which regions represent set C?
Example 15:
• Which regions represent B C?
Example 16:
• Use the Venn diagram to represent each set in roster form.

4 5

6

7 8

9

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

4 5

6

7 8

9

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