1. Lesson 1.2
Set Operations
pp. 6-8

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

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

4. 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

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

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

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}

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

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

10. 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}

11. 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

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

13. 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.

14. 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.

15. 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

16. 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}

17. 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}

18. 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.

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

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

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

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

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

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

25. Homework
p. 8

26. ►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}

27. ►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

28. ►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}

29. ►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

30. ►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

31. ►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,. . .}

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

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

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

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

36. ►B. Exercises 19. (K  L)  (K′  L′) (K'  L') K 9 3 1 12 L 2 10 4 86

37. ►B. Exercises 19. (K  L)  (K′  L′) = {0,5,6,7,11,13,14,15,. . .} K12 L 2 10 4 8 6

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

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

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

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

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

43. 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

44. 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

45. 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