2.3 – Set Operations and Cartesian Products

Intersection of Sets: The intersection of sets A and B is the set of elements common to both A and B.

A B = {x | x A and x B}

{1, 2, 5, 9, 13} {2, 4, 6, 9}

{2, 9}

{a, c, d, g} {l, m, n, o}

{4, 6, 7, 19, 23} {7, 8, 19, 20, 23, 24}

{7, 19, 23}

2.3 – Set Operations and Cartesian Products

Union of Sets: The union of sets A and B is the set of all elements belonging to each set.

A B = {x | x A or x B}

{1, 2, 5, 9, 13} {2, 4, 6, 9}

{1, 2, 4, 5, 6, 9, 13}

{a, c, d, g} {l, m, n, o}

{a, c, d, g, l, m, n, o}

{4, 6, 7, 19, 23} {7, 8, 19, 20, 23, 24}

{4, 6, 7, 8, 19, 20, 23, 24}

2.3 – Set Operations and Cartesian Products

Find each set.

A B

U = {1, 2, 3, 4, 5, 6, 9} A = {1, 2, 3, 4} B = {2, 4, 6} C = {1, 3, 6, 9}

{1, 2, 3, 4, 6}

{6}

{1, 2, 3, 4, 5, 9}

A B A = {5, 6, 9}

B C

C = {2, 4, 5}B = {1, 3, 5, 9)}

B B

2.3 – Set Operations and Cartesian Products

Find each set.

(A C) B

U = {1, 2, 3, 4, 5, 6, 9} A = {1, 2, 3, 4} B = {2, 4, 6} C = {1, 3, 6, 9}

{2, 4, 5, 6, 9}

{5, 9}

A = {5, 6, 9}

A C

C = {2, 4, 5}B = {1, 3, 5, 9)}

{2, 4, 5, 6, 9} B

2.3 – Set Operations and Cartesian Products

Difference of Sets: The difference of sets A and B is the set of all elements belonging set A and not to set B.

A – B = {x | x A and x B}

Note: A – B B – A

{1, 4, 5}

{1, 2, 4, 5, 6, }

U = {1, 2, 3, 4, 5, 6, 7} A = {1, 2, 3, 4, 5, 6} B = {2, 3, 6} C = {3, 5, 7}

A = {7} C = {1, 2, 4, 6}B = {1, 4, 5, 7}

Find each set.

A – B B – A

(A – B) C

2.3 – Set Operations and Cartesian Products

Ordered Pairs: in the ordered pair (a, b), a is the first component and b is the second component. In general, (a, b) (b, a)

True(3, 4) = (5 – 2, 1 + 3)

{3, 4} {4, 3}

False

(4, 7) = (7, 4)

Determine whether each statement is true or false.

False

2.3 – Set Operations and Cartesian Products

Cartesian Product of Sets: Given sets A and B, the Cartesian product represents the set of all ordered pairs from the elements of both sets.

(1, 6),

A = {1, 5, 9}

A B

Find each set.

A B = {(a, b) | a A and b B}

B = {6,7}

(1, 7), (5, 6), (5, 7), (9, 6), (9, 7){ }

(6, 1),

B A

(6, 5), (6, 9), (7, 1), (7, 5), (7, 9){ }

2.3 – Venn Diagrams and SubsetsShading Venn Diagrams:

A B

U

A B

U

A B

U

A B

2.3 – Venn Diagrams and SubsetsShading Venn Diagrams:

A B

U

A B

U

A B

U

A B

2.3 – Venn Diagrams and SubsetsShading Venn Diagrams:

A B

U

A B

U

A B

U

A B

A B in yellow

A

2.3 – Venn Diagrams and SubsetsLocating Elements in a Venn Diagram

Start with A B

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {2, 3, 4, 5, 6} B = {4, 6, 8}

A B

U

6

4

3

5

82

Fill in each subset of U.

Fill in remaining elements of U.

7

9 10

1

2.3 – Venn Diagrams and SubsetsShade a Venn diagram for the given statement.

(A B) CWork with the parentheses. (A B)

A B

CU

2.3 – Venn Diagrams and SubsetsShade a Venn diagram for the given statement.

(A B) CWork with the parentheses. (A B)

BA

CU

Work with the remaining part of the statement.

(A B) C

2.3 – Venn Diagrams and SubsetsShade a Venn diagram for the given statement.

(A B) CWork with the parentheses. (A B)

BA

CU

Work with the remaining part of the statement.

(A B) C

2.4 –Surveys and Cardinal NumbersSurveys and Venn DiagramsFinancial Aid Survey of a Small College (100 sophomores).

49 received Government grants

55 received Private scholarships

43 received College aid

23 received Gov. grants & Pri. scholar.

18 received Gov. grants & College aid

28 received Pri. scholar. & College aid

8 received funds from all three

G

C

P

U

8

(PC) – (GPC) 28 – 8 = 20

20

(GC) – (GPC) 18 – 8 = 10

10

(GP) – (GPC) 23 – 8 = 15

15

43 – (10 + 8 +20) = 55

55 – (15 + 8 + 20) = 12

12

49 – (15 + 8 + 10) = 16

16

100 – (16+15 + 8 + 10+12+20+5) = 14

14

For any two sets A and B,

Cardinal Number Formula for a Region

( ) ( ) ( ).n A B n A n B n A B

Find n(A) if n(AB) = 78, n(AB) = 21, and n(B) = 36.

n(AB) = n(A) + n(B ) – n(AB)

78 = n(A) + 36 – 21

78 = n(A) + 15

63 = n(A)

2.4 –Surveys and Cardinal Numbers