1. 1 Chapter Two Basic Concepts of Set Theory –Symbols and Terminology –Venn Diagrams and Subsets

2. 2 What is a Set? Set is a collection of Objects Objects belonging to the set are called elements of the set, or members of the set.

3. 3 Sets are described in three ways –Word descriptions The set of even counting numbers less than ten –Listing method {2,4,6,8} –Set-builder notation { X| X is an even counting number less than 10}

4. 4 Suppose E is the name for the set of all letters of the alphabet. Then we can write E = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z} We can shorten a listing by using ellipsis points. For example: E = {a,b,c,d,…,x,y,z}

5. 5 Examples List the elements of the set of each of the following: A)The set of counting number between six and thirteen Answer: { 7, 8, 9, 10,11,12}

6. 6 B) List each element of the set { 5,6,7,…10} Answer: Completing the list we get {5,6,7,8,9,10} C) {X | X is a counting number between 6 and 7} Answer: There are no elements – so we write { } or 0

7. 7 Empty or Null Set Empty set is denoted 0 or { } Do not use { 0 } or { 0 } to denote the empty set. Empty set is denoted 0 or { } Do not use { 0 } or { 0 } to denote the empty set.

8. 8 Sets of Numbers Natural or Counting Numbers {1,2,3,4,…} Whole Numbers {0,1,2,3,4,…} Integers {…,-1,0,1,….}

9. 9 Rational Numbers {p/q | p and q are integers and q not equal to 0. (ex. ¾, -7/5, ½ or.55,.67 etc….) Real Numbers {x | x can be written as a decimal } Irrational Numbers {x | x is a real number and x cannot be written as a quotient of integers}

10. 10 Cardinal Numbers The number of elements in a set is called the cardinal number, or cardinality of the set. The symbol n(A) is read “n of A” and represents the cardinality of set A. If elements are repeated in a set, they should not be counted more than once when determining the cardinality of the set. For example, if the set, B = { 1,1,2,2,3,3} there are three distinct elements in the set and n(B) = 3

11. 11 Examples Find the cardinal number of each of the following sets: 1.K = {2,4,8,16} n(K) = 2.M = {0}n(M) = 3.R = { 4,5,…,12,13}n(R) = 4.Empty set 0n(0) =

12. 12 1.K = {2,4,8,16} n(K) = 4 2.M = {0}n(M) = 1 3.R = { 4,5,…,12,13}n(R) = 10 4.Empty set 0n(0) = 0

13. 13 Finite and Infinite Sets If the cardinal number of a set is a whole number or a counting number – then that set is finite set. We can count it. Example: B = { 1,2,3,4,5,6,7,8,9,10} Some sets are so large we cannot count the elements in the set. If the set is so large that its cardinal number is not found among the whole numbers, we call that an infinite set. For example the set of counting numbers is an infinite set. Example B = {1,2,3,4,….}

14. 14 Exercise Review – what are the three common ways to write set notation? Word Description Listing Method Set Builder Notation Now, write the set of all odd counting numbers using a word description, listing method, and set builder notation

15. 15 Set Equality Set A is equal to set B provided the following two conditions are met: 1.Every element of A is an element of B and 2.Every element of B is an element of A.

16. 16 Examples True or False …. {a,b,c,d} = {d,c,b,a} {1,0,1,2,3,} = {0,1,2,3} {4,3,2,-1} = {3,2,4,1} True False

17. 17 Venn Diagrams and Subsets Universe of Discourse –For a problem includes all things under discussion at a given time. Suppose the NOVA Loudon campus considered raising the scores for the Algebra 1 placement exam. The universe of discourse might be all potential students wishing to take Algebra 1 from the Loudon campus.

18. 18 Universal Set In mathematical theory of sets, the universe of discourse is known as the Universal Set. The letter U is usually used for the universal set.

19. 19 Venn Diagrams The universal set is represented by a rectangle, and other sets of interest within the universal set are represented by an oval region, circles, or other shapes.

20. 20 Venn Diagrams U A A’ The entire region bounded by the rectangle represents the Universal Set - U The oval represents the Set A The region inside U and outside the oval is labeled A’ (read A prime) This is the compliment of A Contains elements in U not in A

21. 21 Compliment of a Set For any set A within the universal set U, the complement of A, written A’ is the set of elements of U that are not elements of A. A’ = { X | X E U and X E A }

22. 22 Subset of a Set How do we define the compliment of the universal set, U’. The set U’ is found by selecting all the elements of U that do not belong to U. U A

23. 23 For the universal set U U’ = 0 Next, lets look at the compliment of the empty set, 0 ’. Since 0 ’ = { X | X E U and X E 0 } and set 0 contains no elements, every member of the universal set U satisfies this definition U A

24. 24 So, for every universal set U, 0’ = U Suppose U = {1,2,3,4,5} Let A = {1,2,3} Every element of A is an element of set U Set A is called a subset of set U

25. 25 Subset of a Set Set A is a subset of set B if every element of A is also an element of B. A B Examples

26. 26 Set Equality If A and B are sets, then A = B if A B and B A. Suppose B = { 5,6,7,8} and A= {6,7}. The A is a subset of B, but A is not all of B. A is called a proper subset of B. A B.

27. 27 Proper Subset of a Set Set A is a proper subset of set B if A B and A = B Then A B.

28. 28 Set A is a subset of set B, if every element of set A is also an element of set B. Set A is a subset of Set B, if there are no elements of A that are not also element of B IS the empty set a subset of A or B or both? 0 B –The empty set is a proper subset of every set except itself –Every set (except the empty set) has at least two subsets, the set itself and the empty set.

29. 29 Finding the number of subsets Number of Subsets –The number of subset of a set with n elements is 2 n –Since the value 2 n contains the set itself, we must subtract 1 from this value to obtain the number of proper subsets of a set containing n elements. The number of proper Subset of a set with n elements is 2 n -1

30. 30 Homework Exercises 2.1 Page 54 9 -21 odd, 25, 27, 29, 31, 33, 35, 41 -49 odd, 59 – 66 odd, 67 – 78 odd 2.2 Page 61 7 -41 odd, 43, 45, 49, 52