S ECTION 1.5 Venn Diagrams and Set Operations

O BJECTIVES 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).

K EY T ERMS 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.

K EY T ERMS (C ONTINUED ) 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.

S ETS T AKE ON D IFFERENT F ORMS Disjoint Proper Overlapping Equal A A A A B B B B = Sets with some common elements.

O VERLAPPING S ETS A B Four Regions 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.

N OTE : 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.

E XAMPLE 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}

E XAMPLE 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’

E XAMPLE 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 ∩

E XAMPLE 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

E XAMPLE 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 ∩

S ECTION 1.5 A SSIGNMENTS Classwork: TB pg. 46/ Must write problems and show ALL work to receive credit for this assignment. Homework: Create Engrade Account

E XAMPLE 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 ’

E XAMPLE 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

E XAMPLE 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

E XAMPLE 9: Using the same information from Example 8. ( E H ) ∩ B

K EY T ERMS Difference of Sets: the set of elements that are in B but not in A. This is denoted by B – A.

E XAMPLE 10: U SING S ET D IFFERENCE 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

S ECTION 1.5 A SSIGNMENTS 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.

S ECTION 1.5 CON ’ T Venn Diagrams and Set Operations with Three Sets

T HREE S ETS – 8 R EGIONS I III VII II IV V VI VIII

DeMorgan’s Law (A B)’ = A’ ∩ B’ (A ∩ B)’ = A’ B’

E XAMPLE 11: U = {1, 2, 3, 4, 5, 6, 7} A = {1, 3, 5, 7} B = {1, 2, 3} C = {2, 3, 4, 5, 6}

E XAMPLE 12: U = {1, 2, 3, 4, 5, 6, 7} A = {1, 3, 5, 7} B = {1, 2, 3} C = {2, 3, 4, 5, 6}

E XAMPLE 13: U = {1, 2, 3, 4, 5, 6, 7} A = {1, 3, 5, 7} B = {1, 2, 3} C = {2, 3, 4, 5, 6}

E XAMPLE 14: Which regions represent set C ?

E XAMPLE 15: Which regions represent B C ?

E XAMPLE 16: Use the Venn diagram to represent each set in roster form

E XAMPLE 17: Use the Venn diagram to represent the set in roster form

E XAMPLE 18: Construct a Venn diagram using the following information.

E XAMPLE 19: Determine if the sets are equal using a Venn diagram.

E XAMPLE 20: Determine if the sets are equal using a Venn diagram.

S ECTION 1.5 A SSIGNMENTS 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.