Lesson 1.2 Set Operations pp. 6-8

Objectives: 1. To identify and perform the basic operations on sets. 2. To use Venn diagrams to illustrate the basic operations on sets.

Consider the set U = {x|x is a number from 1-12}. 5 1 3 2 11 9 6 4 12 10 8 7

If A = {3, 6, 9, 12} & B = {x|x is a factor of 12} U 5 A A B 1 3 2 11 9 6 4 12 10 8 7

Find A  B (The union of A & B) This is the set combining all the elements of the given sets.

Find A  B (The union of A & B) U 5 B A 1 3 2 11 9 6 4 12 10 8 7

Example: If A = {1, 2, 4, 7, 11} and B = {x|x is a factor of 28), Find A  B. 1. Ø 2. {1, 2, 4, 7, 14, 28} 3. {1, 2, 4, 7, 11, 14, 28} 4. {1, 2, 4, 7}

Find A  B (The intersection of A & B) The set that contains the elements belonging to both A and B.

Find A  B (The intersection of A & B) U 5 B A 1 3 2 11 9 6 4 12 10 8 7

Example: If A = {1, 2, 4, 7, 11} and B = {x|x is a factor of 28), Find A  B. 1. Ø 2. {1, 2, 4, 7, 14, 28} 3. {1, 2, 4, 7, 11, 14, 28} 4. {1, 2, 4, 7}

If C = {2, 4, 6, 8, 10, 12} & D = {1, 3, 5, 7, 9, 11} U C D 2 1 9 4 10 6 5 3 7 8 12 11

= Ø Find C  D (The sets are disjoint sets) U C D 2 1 9 4 10 6 5 3 7 8 12 11

Operations on a single set are called unary operations. Since two sets are necessary for the operations of union and intersection, they are called binary operations. Operations on a single set are called unary operations.

The last operation (and 1st unary operation) we will look at today is the complement of a set. The complement of a set is the set of all elements in the universal set not in the given set.

If U = {x|x is a number from 1-12} & A = {3, 6, 9, 12}, find A. 5 A 1 3 2 11 9 6 4 12 10 8 7

C′ is shaded in the Venn diagram. C′ = {2, 3, 6} EXAMPLE 1 Draw a diagram of C′. C = {1, 4, 5, 7} and the universal set is the digits from one to seven. U 2 C 7 1 4 5 6 3 C′ is shaded in the Venn diagram. C′ = {2, 3, 6}

Example: If U = {x|x is a whole number from 1-11, inclusive} and A = {1, 2, 4, 7, 11}, find A. 1. Ø 2. {1, 2, 4, 7, 11} 3. {3, 5, 6, 8, 9, 10} 4. {3, 5}

EXAMPLE 2 Find A′  B and (A  B)′. Let A = {1, 2, 3, 4} and B = {1, 3, 5, 7}. Use the same universal set as in example 1.

EXAMPLE 2 Find A′  B A = {1, 2, 3, 4} U A B 1 2 5 3 7 6 4

EXAMPLE 2 Find A′  B A′ U A B 1 2 5 3 7 6 4

EXAMPLE 2 Find A′  B B = {1, 3, 5, 7} U A B 1 2 5 3 7 6 4

EXAMPLE 2 Find A′  B A′  B = {5, 7} U A B 1 2 5 3 7 6 4

EXAMPLE 2 Find (A  B)′ A  B = {1, 3} U A B 1 2 5 3 7 6 4

EXAMPLE 2 Find (A  B)′ (A  B)′ = {2, 4, 5, 6, 7} U A B 1 2 5 3 7 6 4

Homework p. 8

►A. Exercises 3. K  M K  M = {1, 12} Find the following sets and make a Venn diagram to illustrate each operation. 3. K  M K  M = {1, 12}

►A. Exercises Find the following sets and make a Venn diagram to illustrate each operation. 3. K  M = {1, 12} K M 3 1 9 8 12 6 4

►A. Exercises 9. (K  L)  M K  L = {1, 2, 3, 4, 6, 8, 9, 10, 12} Find the following sets and make a Venn diagram to illustrate each operation. 9. (K  L)  M K  L = {1, 2, 3, 4, 6, 8, 9, 10, 12} (K  L)  M = {1, 4, 8, 12}

►A. Exercises Find the following sets and make a Venn diagram to illustrate each operation. 9. (K  L)  M K L M 2 6 3 9 10 12 4 1 8

►A. Exercises 9. (K  L)  M = {1, 4, 8, 12} Find the following sets and make a Venn diagram to illustrate each operation. 9. (K  L)  M = {1, 4, 8, 12} K L M 9 3 6 2 10 4 8 1 12

►B. Exercises 19. (K  L)  (K′  L′) (K  L) = {6} K′ = {0,2,4,5,7,8,10,11,13,14,15,. . .} L′ = {0,1,3,5,7,9,11,12,13,. . .} K′  L′ = {0,5,7,11,13,14,15,. . .} (K  L)  (K′  L′) = {0,5,6,7,11,13,14,15,. . .}

►B. Exercises 19. (K  L)  (K′  L′) K 2 L 6 3 9 10 12 4 1 8

►B. Exercises 19. (K  L)  (K′  L′) K 2 L 6 3 9 10 (K  L) 12 4 1 8

►B. Exercises 19. (K  L)  (K′  L′) K' K 2 L 6 3 9 10 12 4 1 8

►B. Exercises 19. (K  L)  (K′  L′) L' K 2 L 6 3 9 10 12 4 1 8

►B. Exercises 19. (K  L)  (K′  L′) (K'  L') K 9 3 1 12 L 2 10 4 8 6

►B. Exercises 19. (K  L)  (K′  L′) = {0,5,6,7,11,13,14,15,. . .} K 12 L 2 10 4 8 6

■ Cumulative Review A = {1, 3, 8}, B = {1, 3, 9}, C = {3, 8, 9}, D = {3, 9} True/False 21. A  B

■ Cumulative Review A = {1, 3, 8}, B = {1, 3, 9}, C = {3, 8, 9}, D = {3, 9} True/False 22. D  B

■ Cumulative Review A = {1, 3, 8}, B = {1, 3, 9}, C = {3, 8, 9}, D = {3, 9} True/False 23. D  C

■ Cumulative Review A = {1, 3, 8}, B = {1, 3, 9}, C = {3, 8, 9}, D = {3, 9} True/False 24. 9  A  D

■ Cumulative Review A = {1, 3, 8}, B = {1, 3, 9}, C = {3, 8, 9}, D = {3, 9} True/False 25. (A  D)  (B  C)

Find (A  B)  (A  B). (A  B) = {2, 4, 5, 6, 7} U 5 1 3 2 11 9 6 4 12 A B 10 8 7

Find (A  B)  (A  B). (A  B) = {5, 7, 8, 10, 11} U 5 1 3 2 11 9 6 4 12 A B 10 8 7

Find (A  B)  (A  B). (AB)  (AB) = {3,5,6,7,8,10,11,12} U 5 1 3 2 11 9 6 4 12 A B 10 8 7