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 ?

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

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.

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.

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

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 : = ….

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,… }

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

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

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

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

More Notes about Decimal Numbers Repeating Decimal is 1 Repeating Decimal is …. No Repeating Decimal

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

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

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`

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.

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

… 1/2 5/4 -7/3 =

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