1. Discrete Structures – CNS 2300
Text
Discrete Mathematics and Its Applications (5th Edition)
Kenneth H. Rosen
Chapter 1
The Foundations: Logic, Sets, and Functions

2. Section 1.7
Set Operations

3. Unions of Sets
Let A and B be sets. The union of the sets A and B, denoted by , is the set that contains those elements that are either in A or in B, or in both.
union A B

4. Unions of Sets
Let A and B be sets. The union of the sets A and B, denoted by , is the set that contains those elements that are either in A or in B, or in both.
A = {2, 4, 6, 8, 10}
B = {1, 2, 3, 4, 5}
={1, 2, 3, 4, 5, 6, 8, 10}

5. Intersection of Sets
Let A and B be sets. The intersection of the sets A and B, denoted by , is the set containing those elements in both A and B.
intersection A B

6. Intersection of Sets
Let A and B be sets. The intersection of the sets A and B, denoted by , is the set containing those elements in both A and B.
A = {2, 4, 6, 8, 10}
B = {1, 2, 3, 4, 5}
={2, 4}

7. Disjoint Sets
Two sets are called disjoint if their intersection is the empty set.
intersection empty A B

8. Disjoint Sets
Two sets are called disjoint if their intersection is the empty set.
A = {4, 6, 8, 10}
B = {1, 2, 3, 5}
={ }

9. Difference of Sets
Let A and B be sets. The difference of A and B, denoted by A-B, is the set containing those elements that are in A but not in B The difference of A and B is also called the complement of B with respect to A.
difference A B

10. Difference of Sets
Let A and B be sets. The difference of A and B, denoted by A-B, is the set containing those elements that are in A but not in B The difference of A and B is also called the complement of B with respect to A.
A = {2, 4, 6, 8, 10}
B = {1, 2, 3, 4, 5}
A-B ={6, 8, 10}

11. Complement of a Set
Let U be the universal set. The complement of the set A, denoted by A, is the complement of A with respect to U. In other words, the complement of the set A is U - A. A U
complement

12. Complement of a Set
Let U be the universal set. The complement of the set A, denoted by A, is the complement of A with respect to U. In other words, the complement of the set A is U - A.

13. Set Identities x*1 = x Identity laws x+0 = x Domination laws x*0 = 0
Idempotent laws
-(-x)= x
Double negation law

14. Set Identities x+y = y+x Commutative laws x*y = y*x Associative laws
Distributive laws
DeMorgan’s laws

15. Set Identities

16. Proving Equivalences
Use set builder notation and logical equivalences.
Use membership tables.

17. Generalized Unions and Intersections
The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection.
The intersection of a collection of sets is the set that contains those elements that are members of all the sets in the collection.

18. Computer Representation of Sets
Bit vector representation
Representations that contain the actual elements of the set.

19. Problems from the text
Homework will not be collected. However, you should do enough problems to feel comfortable with the concepts. For these sections the following problems are suggested.
Pages 94-96
1-21 odd 41, 43

20. finished