SECTION 1.5Venn 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 and and or.

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 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.

SETS TAKE ON DIFFERENT FORMS

Disjoint

Proper

OverlappingEqual

A

A

A

A

B

B

B

B=Sets with

some common elements.

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.

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.

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}

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’

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 ∩

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

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 ∩

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’ 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’

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

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.

(E H) ∩ 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

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

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’TVenn Diagrams and Set Operations with Three Sets

THREE SETS – 8 REGIONS

IIII

VII

II

IVV

VI

VIII

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}

)( CBA

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

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

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.

)'( CB

1 2 3 10

11

12

4 5

7 8 9

6

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

EXAMPLE 18: Construct a Venn diagram using the following

information.

},,,,,,,,{

},,{

},,,,,{

},,,{

ihgfedcbaU

gfeC

ihfecbB

iheaA

EXAMPLE 19: Determine if the sets are equal using a Venn

diagram.

BA' 'BA

EXAMPLE 20: Determine if the sets are equal using a Venn

diagram.

)''( BA )'( BA

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.