Chapter 1: Sets

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1. Introduction


A set is one of the fundamental concepts of mathematics. The concept of sets is used in many areas of mathematics, such as:



  • i. Relations, ii. Functions, iii. Geometry, iv. Sequences, v. Probability, vi. Statistics


The theory of sets was developed by the German mathematician Georg Cantor (1845–1918).


2. What is a Set ?


A set is a well-defined collection of objects. The objects in a set are called elements or members of the set.


Examples


1.     Odd natural numbers less than 10:{1, 3, 5, 7, 9}


2.     Vowels in the English alphabet:{a, e, i, o, u}


3.     Prime factors of 210:{2, 3, 5, 7}


4.     Solutions of:x² − 5x + 6 = 0 Factorising: (x − 2)(x − 3) = 0, Therefore: x = 2, 3, Solution set = {2, 3}


3. Well-Defined Collection


A collection is called well-defined if we can clearly decide whether an object belongs to the collection or not.


Example


Set of rivers of India


This is well-defined because we can determine whether a particular river is a river of India or not.


For example:



  • Ganga → belongs to the set

  • Nile → does not belong to the set


Not Well-Defined


Collection of the five most famous mathematicians


This is not well-defined because different people may have different opinions about who the five most famous mathematicians are.


4. Important Terms


The following words have the same meaning:


Object = Element = Member


For example: A = {1, 2, 3, 4}


Here, 1, 2, 3 and 4 are elements/members of A.


5. How to Denote Sets


Sets are generally represented by capital letters.


Examples: A, B, C, X, Y, Z


Elements are generally represented by small letters.


Examples: a, b, c, x, y, z


6. Symbol ∈ - Belongs To


The symbol ∈ means “belongs to” or “is an element of.”


Example: A = {1, 2, 3, 4}


Therefore: 2 ∈ A


This means: 2 belongs to A.


7. Symbol ∉ — Does Not Belong To


The symbol ∉ means “does not belong to.”


Example: A = {1, 2, 3, 4}


Therefore: 7 ∉ A


This means: 7 does not belong to A.


Trick : i. ∈ → belongs to,  ii. ∉ → does not belong to


8. Standard Sets


Natural Numbers: N = {1, 2, 3, 4, 5, ...}, N represents the set of natural numbers.


Integers: Z = {..., −3, −2, −1, 0, 1, 2, 3, ...}, Z represents the set of integers.


Rational Numbers: Q represents the set of rational numbers. A rational number can be written in the form: p/q, where q ≠ 0.


Real Numbers: R represents the set of real numbers.


Positive Integers: Z⁺ represents the set of positive integers. Z⁺ = {1, 2, 3, ...}


Positive Rational Numbers: Q⁺ represents the set of positive rational numbers.


Positive Real Numbers: R⁺ represents the set of positive real numbers.


9. Representation of a Set: There are mainly two methods of representing a set: i. Roster or Tabular Form,  ii. Set-Builder Form


10. Roster Form: In roster form, we list all the elements of a set inside curly brackets { }. The elements are separated by commas.


Ex 1: Set of even positive integers less than 7: {2, 4, 6}


Ex 2: Vowels of the English alphabet: {a, e, i, o, u}


Ex 3: Factors of 42: {1, 2, 3, 6, 7, 14, 21, 42}


11. Order of Elements: In a set, the order of elements does not matter.


For example: A = {1, 2, 3, 4} and A = {4, 3, 2, 1} represent the same set.


Trick: Order does NOT matter in a set.


12. Repetition of Elements: An element is generally written only once in a set.


For example, the letters of the word: SCHOOL are: S, C, H, O, O, L But O occurs twice.


Therefore, the set is: {S, C, H, O, L} Not: {S, C, H, O, O, L}


Rule: Repeated elements are counted only once in a set.


13. Set-Builder Form: In set-builder form, we describe the elements of a set by stating their common property.


The general form is: A = {x : x has a particular property}


Here:


: means “such that”


Ex:


Set of vowels: V = {x : x is a vowel in the English alphabet}


Read as: “V is the set of all x such that x is a vowel in the English alphabet.”


14. Example of Set-Builder Form


Consider: A = {4, 5, 6, 7, 8, 9}


These are natural numbers greater than 3 and less than 10.


Therefore: A = {x : x is a natural number and 3 < x < 10}


15. Roster Form → Set-Builder Form


Consider: A = {1, 4, 9, 16, 25, ...}


These are squares of natural numbers.


Therefore: A = {x : x is the square of a natural number}


Alternatively: A = {x : x = n², where n ∈ N}


16. Set-Builder Form → Roster Form


Ex:


Write: A = {x : x is a positive integer and x² < 40} in roster form.


We need positive integers whose squares are less than 40.


1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49 No


Therefore: A = {1, 2, 3, 4, 5, 6}


Note: The largest positive integer satisfying x² < 40 is 6.


17. Solving an Equation and Writing Its Solution Set


Ex: Write the solution set of: x² + x − 2 = 0


Factorise: x² + x − 2 = (x + 2)(x − 1)


Therefore: x + 2 = 0 → x = −2


or


x − 1 = 0 → x = 1


Hence, the solution set is: {1, −2}


18. Matching Roster Form and Set-Builder Form


Consider: {P, R, I, N, C, A, L}


This represents the letters of the word PRINCIPAL.


So it matches: {x : x is a letter of the word PRINCIPAL}


Another example: {1, 2, 3, 6, 9, 18}


These are the positive divisors of 18.


So:


{x : x is a positive integer and is a divisor of 18}


Another: {0}


If:


x + 1 = 1


then: x = 0


So: {x : x is an integer and x + 1 = 1}


matches {0}.


19. Note


1.     Set = Well-defined collection of objects.


2.     Georg Cantor developed the theory of sets.


3.     Elements = Members = Objects


4.     Capital letters are generally used to denote sets.


5.     Small letters are generally used to denote elements.


6.     ∈ means belongs to.


7.     ∉ means does not belong to.


8.     Roster form lists the elements.


9.     Set-builder form describes the common property of elements.


10.                        In roster form, order of elements does not matter.


11.                        Repeated elements are written only once.


12.                        Colon (:) in set-builder notation means “such that.”


20. Practice Set


Q1. Who developed the theory of sets ?


A) Einstein  B) Georg Cantor  C) Newton  D) Pythagoras


Ans: B) Georg Cantor


Q2. Which is a well-defined collection ?


A) Collection of beautiful flowers
B) Collection of intelligent students
C) Collection of prime numbers less than 20
D) Collection of famous people


Ans: C) Collection of prime numbers less than 20


Q3. If A = {2, 4, 6, 8}, which is correct ?


A) 5 ∈ A  B) 6 ∈ A  C) 7 ∈ A  D) 9 ∈ A


Ans: B) 6 ∈ A


Q4. Which symbol means “does not belong to” ?


A) ∈  B) =  C) ∉  D) ⊂


Ans: C) ∉


Q5. The set of vowels in English is:


A) {a, e, i, o, u}  B) {a, b, c, d}  C) {1, 2, 3}  D) {x, y, z}


Ans: A) {a, e, i, o, u}


Q6. Write {1, 4, 9, 16, 25} in set-builder form.


Ans: {x : x is the square of a natural number and x ≤ 25}


Q7. Write {2, 4, 6, 8} in set-builder form.


Ans: {x : x is an even natural number less than 10}


Q8. Write the set of positive integers satisfying x² < 20.


Solution:


1² = 1 , 2² = 4 , 3² = 9 , 4² = 16 , 5² = 25 : No


Ans: {1, 2, 3, 4}


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