1. Survey of Mathematical Ideas Math 100 Chapter 2
John Rosson
Tuesday January 30, 2007

2. Basic Concepts of Set Theory
Symbols and Terminology
Venn Diagrams and Subsets
Set Operations and Cartesian Products
Cardinal Numbers and Surveys
Infinite Sets and Their Cardinalities

3. Power Set The power set of set A, denoted
is the set of all subsets of A. Thus

4. Power Set
Example
In particular, the number of subsets of {1,2,3} is

5. Power Set Theorem: The number of subsets of a finite set A is given by
and the number of proper subsets is given by

6. Power Set Ø 20=1 1-1=0 {a} 1 21=2 2-1=1 {a,b} 2 22=4 4-1=3 {1,2,3} 3
Cardinality
# Subsets
# Proper Subsets Ø
20=1
1-1=0
{a} 1
21=2
2-1=1
{a,b} 2
22=4
4-1=3
{1,2,3} 3
23=8 7
{1,2,c,4,5} 5
25=32 31
{1,2,3,…,100}
100
2100=

7. Complement
The collection of all possible element of sets, either stated or implied, is called the universal set, often denoted U.
For any subset A of the universal set U, the complement of A, denoted
is the set elements of U not in A. Thus

8. Complement
Examples. Let U={a,b,c,d,e,f,g,h,i,j,k,l}, A={a,b,c,d}. Let C  D.

9. Venn Diagrams
A Venn diagram is a pictorial representation of sets and their various relations and operation. The first picture below represents the universal set U, a set A, and the complement of A. The second represents the relation

10. Numbers as Sets
All mathematical objects can be defined in terms of sets. The example below indicates how one might define the first five whole numbers as sets.

11. Intersection The intersection of sets A and B, denoted
is the set of elements common to both A and B.

12. Intersection
Let A and B be sets with A  B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set
Sets with empty intersection are called disjoint. Thus, every set is disjoint from its complement.

13. Union The union of sets A and B, denoted
is the set of elements belonging to either of the sets.

14. Union
Let A and B be sets with A  B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set

15. Difference The difference of sets A and B, denoted
is the set of elements belonging to set A but not to set B.

16. Difference
Let A and B be sets with A  B. Let C = {1,2,3} and D = {3,a,b}. Let U be the universal set

17. Ordered Pairs
The ordered pair of with first component a and second component b, denoted
is defined to be the set
Thus,
Note that for ordered pairs, order is important. So
In particular,

18. Cartesian Product The Cartesian product of sets A and B, denoted
is the set

19. Cartesian Product Let C = {1,2,3} and D = {3,a,b}.
In general, for sets A and B:
So in the example

20. Operations on Sets
Operations on sets can be combined. Let A={a,b}, B={b,c}, C={c,d}, D={b,d} and E={a,c}. Calculate in list form.
Working from the inside out

21. Venn Diagrams
Here is the previous set calculation as a Venn diagram. The is no adequate Venn diagram for the Cartesian product. a b c A B b c d C c b d D

22. De Morgan’s Laws
For and sets A and B, the complement of their intersection is the union of their complements,
and the complement of their unions is the intersection of their complements.

23. De Morgan’s Laws
The set will be all the blue not in A and not in B.

24. Assignments 2.4, 2.5, 3.1 Read Section 2.4 Due February 1
Exercises p. 79
1, 3, 5, 7, 9, 17, 19, 25, and 27.
Read Section 2.5
Due February 6
Exercises p. 88
1-6, 7, 9, 11, 13, 14, 15, 24, 29, 32, 37, 38, 39, 40, 43.
Read Section 3.1
Due February 8
Exercises p. 99
1-9, 39-47, 49-53, 57-74