CHAPTER 1 - Sets and Intervals

Learning Objectives: Use the arithmetic properties of subsets of integers, rational, irrational and real numbers, including closure properties of the four basic arithmetic operations where applicable.

Sets A set is a well-defined collection of objects or numbers. These objects are called the elements of the set. Examples The collection of natural numbers. The group of students in a particular section. Vowels of English Alphabet.

Set Notation Example S= {a, b, c, …………………} We use the symbol ∈ to state if an object is a member of a set. We read it as ‘belongs to’. Since a is an element of the set S, we write, a ∈𝑺 If the object or number is not an element of the set, we use the symbol Since 1 is not an element of the set S, we may write as 1 𝑆

Example Let P = {1 , 3 , 5 , 7 ,……………………….}, the set of all odd natural numbers. State if the following statements are TRUE or FALSE: a) 1 is an element of P. TRUE b) -5 is an element of P. FALSE c) 0 is not an element of P. TRUE d) 6 𝝐 P FALSE FALSE e) 9 𝝐 P

Representation of sets: 1. Roster form (Listing of elements) In this form elements of the set are listed within the pair of brackets { } and are separated by commas.  Example Let A denotes the set of natural numbers less than 6 A = {1, 2, 3, 4, 5} Roster form b) The set of odd numbers less than 8 B = {1, 3, 5, 7} Roster form

2. Set builder form In this form, instead of listing all the elements, we write the general rule satisfied by the elements of the set Example a) Set of integers between 3 and 8 P = {𝑥 : 𝑥 𝝐 Z , 3< 𝑥 <8 } Set builder form b) Set of whole numbers greater than 2 M = {𝑥 : 𝑥 𝝐 W , 𝑥 >2 } Set builder form

Empty set If there is no elements in the set its called an empty set. It is denoted by { } OR ∅ Example The set of all integers between 1 and 2 is empty or ∅

2) It doesn’t matter what order the elements are present in a set. Note: 1) Elements of a set need not be repeated. a) {1, 2, 3, 3, 4, 5} Here, the element 3 is not required to be repeated. {1, 2, 3, 4, 5} will do. b) {Math, English, comp, Math } Here, the element Math is not required to be repeated. {Math, English, comp} will do. 2) It doesn’t matter what order the elements are present in a set. i.e, {1, 2, 3, 4} = {4, 2, 1, 3}

Operation on Sets: 1. Union of two sets: The union of two sets A and B, denoted by A∪B, is the collection of all elements of A or B or both. That is, if x∈𝐴 or x∈𝐵, then x∈𝐴∪𝐵. A∪𝐵={𝑥:x∈𝐴 or x∈𝐵} 

Shaded part in the following diagram represents A∪B

Example If A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {6, 8, 9, 10} Then find A ∪ B Solution: A∪ B = {1 , 2, 3, 4, 5, 6, 7 , 8 , 9, 10}

Example Let W denotes the set of all whole numbers and N, the set of all natural numbers. Then find W ∪ N Solution We know that, W = {0, 1, 2, 3, 4, …..} and N = {1, 2, 3, 4, …..} Thus, W ∪N = {0, 1, 2, 3, 4, …..} That is, W ∪N = W

Operation on Sets: 2. Intersection of two sets: The intersection of two sets A and B, denoted by A∩B, is the collection of all common elements of both A and B. That is, if x∈𝐴 and x∈𝐵, then x∈𝐴∩𝐵. A∩𝐵={𝑥:x∈𝐴 and x∈𝐵} 

Shaded part in the following diagram represents A∩B

Example If A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {6, 8, 9, 10} Then find A ∩ B Solution: A ∩ B = {6, 8}

Example Let Z denotes the set of all integers and N, the set of all natural numbers. Then find Z ∩ N Solution We know that, Z = {…, -3, -2, -1, 0, 1, 2, 3, 4, …..} and N = {1, 2, 3, 4, …..} Thus, Z ∩ N = {1, 2, 3, 4, …..} That is, Z ∩ N = N

Intervals Any connected portion of a number line is said to be an interval. There are different types of intervals such as open intervals, closed intervals and half(semi) open intervals depending on the inclusion or exclusion of the end points.

Different types of Intervals Inequality Graph Open interval ( a , b ) 𝒂<𝒙<𝒃 a b Closed interval [ a , b ] 𝒂≤𝒙≤𝒃 a b Half open (or half closed) ( a , b ] 𝒂<𝒙≤𝒃 a b [ a , b) 𝒂≤𝒙<𝒃 a b

Infinite End points Interval Inequality Graph 𝒂,∞ 𝒙> 𝒂 -∞ a ∞ −∞,𝒂 𝒙<𝒂 ∞ -∞ a [𝒂, ∞ ) 𝒙≥𝒂 -∞ a ∞ (−∞ , 𝒂] 𝒙≤𝒂 ∞ a -∞

Absolute value of a number: The absolute value of a number is the distance of that number from zero. It is always positive. Absolute value of a number 𝑎 is denoted by 𝑎 . Examples a) 5 = 5 b) −7 = 7 c) −10 + −10 −[(−10)×3] Solution = 10 −10−(−30) =10 −10+30 =30

+ − Example 𝜋≈3.14 a) 𝜋−2 𝜋−2 NO CHANGE Solution: ≈3.14−2 𝜋−2 =𝜋−2 ≈1.14 + b) 2−𝜋 Solution: CHANGE THE ORDER 2−𝜋 2−𝜋 =−(2−𝜋) Multiply by − ≈2−3.14 =− 2+𝜋 − ≈−1.14 or, = 𝜋− 2

Distance between points on the number line d(a, b) 𝒂 𝒃 If 𝒂 and 𝒃 are real numbers, then the distance between the points 𝒂 and 𝒃 on the real line is given by, 𝒅 𝒂,𝒃 = 𝒃−𝒂

Examples Find the distance between the given numbers: a) 2 and 5 b) - 3 and 4 Solution: d = 5−2 d = 3 d = 3units Solution: d = 4−(−3) d = 7 d = 7 units