Sets, Relations & Functions (Part-1)

 

SETS

 Lesson 1

In this lesson, we will discuss sets and different types of sets. First, let us understand what a set is.

A set is a well-defined collection of distinct objects or elements.

For example: (i) When we refer to living beings (such as human beings, animals, plants) or non-living things (such as numbers, letters, symbols, etc.), we are essentially referring to some collection.

(ii) To call a collection “well-defined,” it must be clear whether a particular object belongs to that collection or not.

For example:

The collection of all natural numbers less than `10` is well-defined, because it is clear that `8` belongs to the collection, whereas `12` does not.

The collection of all students of a particular school is well-defined, because it is clear who is a student and who is not.

The collection of all capital cities in India of different states is well-defined, because each city is either a capital or not.

However, if we consider the collection “the set of all beautiful cities in India" it is not well-defined, because beauty is subjective—what one person considers beautiful, another may not.

Similarly, if we consider “the set of interesting books,” this is also not well-defined, because what one person finds interesting may not be interesting to someone else.

Therefore, vague or ambiguous collections are not considered sets.

(iii) Mutual distinctness means that in a set, no object can appear more than once.

For example: If the set of natural numbers less than `5` is written as `{1, 1, 2, 3, 4}`, it is not considered a proper set, although it is well defined, because 1 appears twice.

The correct set is `{1, 2, 3, 4}`, where each element is listed only once. Repetition of elements is not allowed in a set.

Also, the order in which elements are written is not important. `{1, 2, 3, 4}` and `{4, 3, 2, 1}` represent the same set.

In English, sets are usually denoted by capital letters such as `A, B, C, …` and the elements by small letters such as `a, b, c, … .`

A well-defined collection of distinct objects is called a set. Any object contained in a set is called an element or member of the set. If `a` is an element of a set `A`, then we say “`a` belongs to set `A`” or “`a` is a member of `A`.”

If an element `a` belongs to a set `A`, we write it as `a \in A` and (`a \in A`) read it as “a belongs to A.” If an element `c` does not belong to a set `B`, then we write `c \notin B` and (`c \notin B`) read it as “c does not belong to B.”

In general, there are two ways to represent sets: (i) Listing all the elements of the set explicitly in a curly bracket separated by commas. This method of representation of set is known as Roster Method or Tabular Method (ii) Describing the common property that all elements of the set share. This method of representation of set is known as Set builder Method or Property Method. 

For example, if we consider the set of natural numbers from 1 to 5, we can represent it in two ways:

`A={1,2,3,4,5}` or, `A = \{ x |\text{x is a natural number and } 1 \leq x \leq 5 \}` 

 In the second representation, the symbol “|” is read as “such that.” So, `A` represents the set of all `x` such that `x` is a natural number from 1 to 5.

Example (1): (i) If Z denotes the set of integers, then we can write

`Z = \{ 0, \pm 1, \pm 2, \pm 3, \ldots \}` or `Z = \{ x |x \text{is any integer} \}`

In this case, we say `0\in Z`,  `−1\in Z`, `1 \in Z`; but `\frac{1}{2} \notin Z`, because `\frac{1}{2}` is not an integer.

(ii) If A denotes the set of the seven days of the week, then we can write

`A = \{ \text{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} \}` or
`A=\{x|x\text{ is a day of the week}\}`

(iii) If `B` denotes the set of vowels in the English alphabet, then we can write

`B = \{ a, e, i, o, u \}` or `B = \{ x|x\text{ is a vowel in the English alphabet} \}`

If there are no special instructions, in this chapter, sets will usually be represented by the following symbols and its meaning mentioned besides the symbol.

Some Standard Sets and Their Symbols:

Symbol                          Meaning

N                               Set of natural numbers

Z                                    Set of integers

Z⁺                                   Set of positive integers

Q                                    Set of rational numbers

Q⁺                                   Set of positive rational numbers

R                                    Set of real numbers

R⁺                                   Set of positive real numbers

C                                    Set of complex numbers

In set-builder notation: “: or |” means “such that,” and the collection within braces represents the set.

Finite and Infinite Sets

If a set has a finite number of elements, it is called a finite set. If a set has an infinite number of elements, it is called an infinite set. for example, `A=\{1,2,3\}` is a finite set; but the set of natural numbers `N=\{1,2,3,...\}` is an infinite set. 

Example (2): Which of the following sets are finite or infinite

(a)    The set of a months of year

(b)    `A=\{x|x \text{are roots of the equation }x^3-3x^2-x+3=0\}`

(c)     The set of positive integers greater than `100`

(d)    `A=\{x\inR:0<x<1\}`

Solution (a): In this case, the set is finite, because the number of months in a year is 12.

(b):  Here, the roots of the equation `x^3-3x^2-x+3=0` are `1,-1,3` i.e., `A=\{1,-1,3\}`. Hence the set is finite.

(c): In this case, the set is infinite because there are infinite numbers of positive integers greater than `100`. Hence the set of positive integers greater than `100` is an infinite set

(d): In this case also the set `A` is infinite because there are infinite numbers of real numbers between `0` and `1` Hence The set `A` is an infinite set.

Empty Set or Null Set or Void Set: A set which contains no element is called null set. It is denoted by `\phi`. for example, `A=\{x|x\ text {is the root of }x^2+1=0,x\inR\}`. Here `x^2+1=0` gives `x^2=-1` but there is no real number whose square is negative. Hence the set `A` contains no elements. So, the set A is null set. 

Remark (a) Null set is unique. So, we write 'The Null Set' not 'A Null Set' 

(b) The null set is a finite set. (c) The set `\{0\}` is not null set since the set has one element which is `0`. 

Singleton Set: A set which contains a single element only is called singleton set. 
Example (a): The set of even prime number `\{2\}` is a singleton set.
(b): `A=\{x:2x-3=0,x\inR\}=\{\frac{3}{2}\}`is a singleton set.

Equal Sets: Two sets `A` and `B` are said to be equal i.e., `A=B` if any `x\inA\impliesx\in B` and any `y\inB\impliesy\inA`.

Example: The set `A=\{x:x  \text{is a letter of the word FOLLOW}\}` and `B=\{y:y  \text{is a letter in the word WOLF}\}` are equal sets because `A=\{F,L,L,O,O,W\}=\{F,L,O,W\}` and `B=\{W,O,L,F\}` i.e., the elements of `A` are also the elements of `B` and elements of `B` are also the elements of `A`. Hence any `x\inA\impliesx\in B` and any `y\inB\impliesy\inA` is satisfied.

Solved Example

1. Which of the following are sets? Justify your answer.

(a) The collection of all the months of a year beginning with the letter J.

(b) The collection of ten most talented writers of India.

(c) The collection of all natural numbers less than `100`.

(d) The collection of novels written by the writer Sunil Gangopadhya.

Solution (a): The collection of the months of a year beginning with the letter J form a set as the month beginning with the letter J are distinct from one another and it is possible to distinguish the months in the set from those not in the set.

(b): The collection of `10` most talented writers of India do not form a set. The word 'talented' is not well-defined, because the attribute 'talented` is subjective—what one person considers talented, another may not.

(c): The collection of all natural numbers less than `100` form a set as the natural numbers less than`100`are distinct from one another and it is possible to distinguish the natural numbers in the set from those not in the set. 

(d): The collection of novels written by the writer Sunil Gangopadhya form a set as the novels written by Sunil Gangopadhya are distinct from one another and it is possible to distinguish the novels written by Sunil Gangopadhya from those are not written by Sunil Gangopadgya.  

2. Write the following sets in roster form

(a) `A=\{x\inN:5<x<10\}`   (b) `B=\{x|x  \text{is vowels of English alphabets}\}`

(c) `C=\{x: x \text{is a two dight natural number such that the sum of the digits is 8}\}`

(d) `D=\{x:x \text{ is the letters of the word TRIGONOMETRY}\}`

Solution (a): Natural numbers between `5` and `10` are `6,7,8,9`. So, the given set in roster form is `A=\{6,7,8,9\}`.

(b): The vowels in English alphabets are `a,e,i,o,u`. So, the given set in roster form is `B=\{a,e,i,o,u\}`.

(c): The letters of the word TRIGONOMETRY are `E,G,I,M,N,O,R,T,Y`. Here the letters `T,R,O` are repeated. Thus, the given set in roster form is `D=\{E,G,I,M,N,O,R,T,Y\}`

3. Write the following sets in the set builder form:

(a) `A=\{1,2,3,4,5,6\}`   (b) `B=\{2,4,6,8,10\}`   (c)  `C=\{0,1,4,9,16,...}`

(d) `D=\{a,e,i,o,u\}`

Solution (a): `A=\{x:1leqxleq6,x\inN\}`,  (b): `B=\{x:x=2n,1leqnleq5  \text{and}  n\inN\}`
(c): `C=\{x:x=n^2,n\inZ\}`,  (d): `D=\{x: x  \text{is vowel of English alphabet}\}` 

4. Which of the following sets are finite and which are infinite? 

(a) `A=\{x:2<x<3,x\inR\}`   (b) `B=\{1,2,3,...,100\}`  (c) `C=\{x:x=3n,n\inN\}`

(d) Family of circls in a plan having a fixed centre

Solution (a): The set of real numbers lying between `2` and `3` contains an uncountable number of elements. So it is an infinite set.

(b): The set `B` contains first `100` natural numbers, thus the elements of the set are countable. Hence the set is finite.

(c): The set `C=\{x:x=3n,n\inN\}=\{3,6,9,12,...\}`contains the natural numbers which are divisible by `3` and as natural numbers are uncountable hence it is an infinite set.

(d): There are uncountable number of circles in a plane having fixed centre. So, the number of elements in the set of circles in a plane is not finite. Hence, it is an infinite set.

5. Which of the following sets are the null sets?

(a) `A=\{x|x  \text{is prime number which is divisible by 2}\}`

(b) `B=\{x|x  \text{is even prime number greater than 2}\}`

(c)  The set of point of intersections of two parallel straight lines.

(d) `X=\{x|x  \text{is a positive proper fraction and} x>\frac{1}{x}}`

Solution (a): The set `A` is the set of prime numbers divisible by 2 i.e., even prime number. There is the only one even prime number which is `2`. Thus, the set contains only one element. So, it is a singleton set not null set.  

(b): The set `B` contains elements which are even prime number greater than `2`, but there is only one even prime number 2. So, there are no even prime numbers greater than `2`. Hence, the set contains no elements, i.e., the set is null set.

(c): We know that two parallel lines can never intersects each other in reality. So, there is no point of intersections. Thus, the set contains no elements, i.e., the set in a null set.

(d): As `x` is a positive proper fraction, `\therefore0<x<1\implies\frac{1}{x}>1\impliesx<\frac{1}{x}`.  

Thus, there is no positive proper fraction `x` which is greater than `\frac{1}{x}` i.e., `x>\frac{1}{x}`. Hence, the set `X` is a null set.  

6. Are the following pair of sets are equal? Give reasons.

(a) `A=\{2,6\}`,  `B=\{x|x  \text{is a solution of }x^2+5x+6=0\}`.

(b) `A=\{x:x  \text{is a letter of the word}  LOAKT\}`,  `B=\{x|y  \text{is a letter in the word}  KOLKATA\}` 

Solution (a): We have, `A=\{2,3\}` and `B=\{x|x  \text{is a solution of}  x^2+5x+6=0\}=\{-2,-3\}`. Clearly, `x\inA\impliesx\inB` and `y\inB\impliesy\inA` is not happned. Hence `A\neB`. 

[Remark: Here, we see that both the sets `A` and `B` have equal number of elements. These types of sets are called equivalent set. Hence `A` and `B` are equivalent set not equal set ]

(b): We have, `A=\{L,O,A,K,T\}`, `B={K,O,L,K,A,T,A\}` . Here `A` and `B` are equal sets as repetition of elements in a set do not change a set. Thus, `A=\{L,O,A,K,T\}=B`.

Exercise-1(A) 

(MCQ)

A. Chose thee correct Answer 

1. Which one of the following is not a set
(a) The collection of all boys in your class
(b) The collection of natural numbers less than `100`
(c) A collection of novels written by the writer Atin Bandopadhya
(d) The collection of five most talented actors in west Bengal

2. The roster form of the set of natural numbers less than 6 is
(a) `\{-1,-3,-5,1,3,5\}`
(b) `\{2,3,4,5\}`
(c) `\{1,2,3,4,5\}`
(d) `\{0,1,2,3,4,5,-1,-2,-3,-4,-5}`.

3. The set-builder form of the set `\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7}\}`
(a) `\{x:x=\frac{n}{n+1},  \text{where n is a real number and}  1\leqn\leq6\}` 
(b) `\{x:x=\frac{n}{n+1},  \text{where n is a natural number and}  1\leqn\leq6\}`
(c) `\{x:x=\frac{n}{n+1},  \text{where n is an integer and}  1\leqn\leq6\}`
(d) `\{x:x=\frac{n}{n+1},  \text{where n is a whole number}  1\leqn\leq6\}`.   

4. Match each of the set on the left in the roster form with the same set on the right described in set-builder form

(i)    `\{1,2,3,6\}`                                 (a) `\{x:x  \text{is a prime number and a divisor of }6\}`

(ii)    `\{2,3\}`                                         (b) `\{x:x  \text{is an odd natural number less than}10\}`

(iii)    `\{M,A,T,H,E,I,C,S\}` (c) `\{x:x  \text{is natural number and divisor of 6}\}` 

(iv)    `\{1,3,5,7,9\}`                         (d) `{\x:x  \text{is a letter of the word MATHEMATICS}\}` 

Which one of the above is correct?

(a) `(i)\leftrightarrow(c)`, `(ii)\leftrightarrow(a)`, `(iii)\leftrightarrow(d)`, `(iv)\leftrightarrow(b)` 
(b) `(i)\leftrightarrow(a)`, `(ii)\leftrightarrow(c)`, `(iii)\leftrightarrow(b)`, `(iv)\leftrightarrow(d)` 
(c) `(i)\leftrightarrow(b)`, `(ii)\leftrightarrow(c)`, `(iii)\leftrightarrow(d)`, `(iv)\leftrightarrow(a)`
(d) none of these.  

5. Which of the following set is not finite?
(a) `\{x\inN:5<x<10\}`
(b) `\{x:x  \text{prime numbers lying between 4 and 16}\}`
(c) `\{x\inR:2<x<3\}`
(d) `\{x:x  \text{is vowels in the word MOTHER}\}`

6. Which of the following set is not a null set?
(a) `\{x:x  \text {is a prime number divisible by 2}\}`
(b) `\{x\inN:x<3 and x>5\}`
(c) Set of common multipliers of `4` and `15` except `1`
(d) Set of point of intersections of two parallel straight lines.

7. Which one of the following pairs of sets are equal?
(a) `A=\{x:x  \text{is letters of the word CATARACT}\}`, `B=\{X:X  \text{is the letters of the word TRACT}\}`
(b) `X=\{2,3,5,7,11\}`,  `Y=\{x:x  \text{is prime numbers less than11}\}`
(c) `S=\{1,\omega,\omega^2\}`,  `T=\{x:x  \text{is the complex roots of the equation}x^3=1\}`
(d) `P=\{x|x  \text{is the digits of the number}15212573\}`,  `Q=\{x:x  \text{is the digits of the number}23217537\}`.