1. The Real Numbers 1.1 Sets A set is a collection of objects, symbols, or numbers called elements. Example 1 is a set containing the first three counting numbers. 1, 2, and 3 are elements of the set. Example 2 is a set containing the the vowel letters in English language Question: What are the elements of this set? Answer: The elements are: a, e, i, o, and u. Class Exercise Let D = { x / x is a day of the week }. What are the elements of D ?

2. Using the symbols is used to Any object or symbol that is contained in a set is called an element, or a member, of the set. The symbol indicate that an object is an element of the set. Example 1 3Set A = { 1, 2, 3, 4 }. Example 2 January D = { x / x is a day of the week }. Class Excercise Complete each statement with the symbols If B = { 1, 3, 5, a, c } i) a Bii) 2 B iii) c B

3. Equal Sets Two sets are equal if they contain the same elements. Example 1 Let A = { a, b, c, d } and B = { a, d, b, c }. Since A and B have the same elements, then they are equal. We Write A = B Ordered Sets If the elements of a set can be ordered and we wish to indicate that the set continues as described, we use an ellipses, three dots that mean “ and so on”. Example 1 The set { a, b, c, …, z } represents the entire alphabet Example 2 The set { 1, 2, 3, …, 100 } represents the counting numbers from 1 till 100.

4. More Examples Example 3 The set{ 1, 3, 5, … } represents the positive odd numbers Example 4 The set{ 2, 4, 6, … } represents the positive Even numbers Finite or Infinite Sets A set that has a specific number of elements is said to be finite, otherwise, it is infinite. Example 1 The set A = { 1, 2, 3} is finite. Example 2 The set B = { 1, 2, 3,…, 10} is finite. Example 3 The set C= {2, 3,4,…} is infinite.

5. More Examples Example 4 The Set N = { 1, 2, 3, … } = Set of Natural numbers and it is infinite The Set W = { 0, 1, 2, 3, … } = Set of Whole numbers and it is infinite Example 5 Example 6 The Set Z = { …,-3, -2, -1,0, 1, 2, 3, … } = Set of Integer numbers and it is infinite

6. Important Notes Every element in N is in W, and every element in W is in Z. Class Exercise Complete each statement with the symbols i) 0 N ii) 0 Z iii) 0 W iv) 1 Z v) -1 N vi) -1 W vii) -1 Z viii) 7 Z ix) Z x) W Note : =3.141828….

7. Venn Diagram 1 Set of Natural Numbers N = { 1,2,3,…} Set of Whole Numbers W = {0,1,2,3,…} Set of Integers Z = {…, -3, -2, -1, 0, 1, 2, 3,… }

8. Rational Numbers are considered as Rational Numbers Numbers as ½ 0.34 5 1.333… 5.2323… -1.5 ¾ -3 Because we cannot list the rational numbers in any meaningful fashion, we define the elements of that set as:

9. Examples of Rational numbers i) 1/2 Q ii) 0 Q iii) 0.34 Q iv) -1 Q iv) 1.333 Q

10. Venn Diagram 2 Set of Natural Numbers N = { 1,2,3,…} Set of Whole Numbers W = {0,1,2,3,…} Set of Integers Z = {…, -3, -2, -1, 0, 1, 2, 3,… } Set of Rational Numbers Q

11. Important Notes About Rational Numbers The numbers 3.5 3.111… 2.6565… 3.141828…. Are decimal numbers Not all Decimal numbers are rational numbers 3.5 Q 3.111… Q 2.6565…… Q 3.141828… Q Terminating Decimal Repeating Decimals

12. More Notes about Decimal Numbers Repeating Decimal is 1 Repeating Decimal is 65 3.141828…. No Repeating Decimal

13. Class Participation About Rational Numbers…. Class Exercise Complete the following table with Yes or No NumberNWZQ 0NoYes 9 -4No Yes 3.8No Yes 2.546No Yes 8.222…No Yes NO

14. More Class Practice Class Exercise From the set List the elements in N,Z,Q How about the elements

15. Irrational Numbers If a number is not rational, then it is irrational Q = Set of Rational Numbers Q` = Set of Irrational Numbers Example 1 Q Q` Q Class Exercise Check whether these numbers are rational Q, or Irrational Q`

16. Real Numbers The set of real numbers is the union of the sets of rational numbers and irrational numbers. Rational Numbers Q Irrational Numbers Q` ( Not Q ) Real Numbers R All Numbers in N, Z,Q, and Q` are real numbers.

17. Real Line (Numbered Line ) Numbered Line ( Real Line ) Class Exercise On the number line provided, graph the points named by each set

18. … 1/2 5/4 -7/3 =- 2.333

19. Home Work Assignment Do all the home work exercises in the syllabus