Set Theory
3.0 Types of set
3.0 Types of set
Null set
A set which does not contain any element is called a null set and is denoted by $$\emptyset $$
It is also called an empty set.
Illustration 6. Which of the following are null set. Justify the answer.
a. The set of vowels in the word RHYTHM
b. $\{ x:5 < x < 6,\,x \in N\} $
c. The set of natural numbers that are divisible by $13$
d. $\{ x:{x^2} < 0,\,x \in R\} $
e. The set of letters that form the word DEDICATION
f. $\{ x:x = {n^n},\,n \in N\} $
Solution:
| SET | CLASSIFICATION |
| a. The set of vowels in the word RHYTHM | The word RHYTHM does not contain any vowel in it. Hence the set of vowels in the word is null. Thus, this is a null set. |
| b. $\{ x:5 < x < 6,\,x \in N\} $ | As $x$ is a natural number, it can take only positive integer values. Given that it lies between $5$ and $6$. But, there are only decimals between the numbers. Hence, there is no value of $x$ such that $5 < x < 6$ and $x$ is natural number. Thus, this set is a null set. |
| c. The set of natural numbers that are divisible by $13$ | All multiples of $13$ are divisible by $13$. There are many such numbers. Hence, this set is not a null set. |
| d. $\{ x:{x^2} < 0,\,x \in R\} $ | Square of any number is a positive number. It can never be negative. Thus, this set is a null set. |
| e. The set of letters that form the word DEDICATION | The word DEDICATION is made of A,C,D,E,I,N,O,T. Hence this set is not a null set. |
| f. $\{ x:x = {n^n},\,n \in N\} $ | This set has as many number of elements as the number of positive integers. Hence, this set is not a null set. |
Singleton set
A set which contains only one element is called a singleton set.
Illustration 7. Which of the following are singleton set. Justify the answer.
a. The set of vowels in the word STRENGTH
b. $\{ x:{x^2} = 4,\,x \in Z\} $
c. $\{ x:5 < x < 7,\,x \in N\} $
d. The number of natural numbers that are divisible by $2$
e. The number of alphabets in the English language
f. $\{ x:{x^2} < 4,\,x \in N\} $
Solution:
| SET | CLASSIFICATION |
| a. The set of vowels in the word STRENGTH | The vowels in the word STRENGTH is E only. This set contains only one element. Hence, this is a singleton set. |
| b. $\{ x:{x^2} = 4,\,x \in Z\} $ | Given ${x^2} = 4$ and $x \in Z$. Thus, $x$ can take the values $ - 2,2$. This set has two elements and hence is not a singleton set. |
| c. $\{ x:5 < x < 7,\,x \in N\} $ | The only number that satisfies the given condition is $6$. Hence, this is a singleton set. |
| d. The number of natural numbers that are divisible by $2$ | All even numbers are divisible $2$. There are infinite number of elements in this set. Hence, this is not a singleton set. |
| e. The number of alphabets in the English language | There are $26$ alphabets in the English language i.e. $26$ elements in the set. Hence this is not a singleton set. |
| f. $\{ x:{x^2} < 4,\,x \in N\} $ | Given that ${x^2} < 4$ and $x \in N$. Thus, $x < 2$ i.e. $x = 1$. There is one element in this set. Hence, this is a singleton set. |
Finite set
A set $A$ is said to be a finite set if it contains only finite number of elements. The number of elements can be found out and is definite.
Illustration 8. Write the elements of the following sets if they are finite.
a. $\{ x:x < 7,\,x \in N\} $
b. The set of all vowels forming the word AFFECTION
c. $\{ x:{x^2} < 4,\,x \in N\} $
d. The number of alphabets in the English language
e. $\{ x:{x^2} > 1,\,x \in N\} $
f. The set of all numbers that lie between the even prime number and the first two digit number.
Solution:
| SET | ELEMENTS |
| a. $\{ x:x < 7,\,x \in N\} $ | Elements of the given finite set are, $1,2,3,4,5,6$ |
| b. The set of all vowels forming the word AFFECTION | Elements of the given finite set are, $A,E,I,O$ |
| c. $\{ x:{x^2} < 4,\,x \in Z\} $ | Elements of the given finite set are, $1,-1$ |
| d. The set of alphabets in the English language | Elements of the given finite set are, $A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z$ |
| e. $\{ x:{x^2} > 1,\,x \in N\} $ | There are infinite number of elements for this set. |
| f. The set of all real numbers that lie between the even prime number and the first two digit number. | The set is defined as $2 < x < 10$. And since $x$ is a real number, there are infinite number of numbers between them. |
Infinite set
A set $A$ is said to be an infinite set if it contains infinite number of elements i.e. total number of elements in the set is unknown.
Illustration 9. Classify the following as finite or infinite. Justify the answer.
a. Set of concentric circles in a plane
b. Set of all two digit natural numbers
c. Set of all numbers that are prime factors of $178$
d. Set of points on a plane
e. Set of all numbers greater than $1000$
f. Set of people living in a city
Solution:
| SET | CLASSIFICATION |
| a. Set of concentric circles in a plane | There are infinite number of concentric circles that can be drawn with a single point as center. The radius can take any value. Hence, this is an infinite set. |
| b. Set of all two digit natural numbers | The first two digit number is $10$ and the last two digit number is $99$. This is definite, and can be counted. Hence, this is a finite set. |
| c. Set of all numbers that are prime factors of $178$ | The prime factors of $178$ can be noted and this set has finite number of elements. Thus, this is a finite set. |
| d. Set of points on a plane | There are infinite number of points on a plane. Hence, this set is infinite. |
| e. Set of all numbers greater than $1000$ | Here, $x>1000$, i.e. $x$ can take the value from $1000$ to infinity. Hence, this is an infinite set. |
| f. Set of people living in a city | The set of people living in a city can be counted by simply taking the census. Hence, this is a finite set. |
Equal Sets
Two sets are said to be equal, if every element of one set is in the other set and vice versa.
The elements are same in both the sets.
$$\begin{equation} \begin{aligned} \forall \;x \in A\;\;if\;x \in B\;and\;n(A) = n(B)\; \\ \Rightarrow A = B \\\end{aligned} \end{equation} $$
Equivalent Sets
Two sets $A$ and $B$ are equivalent if number of elements are the same in both sets, and each element of set $A$ can be paired with unique elements of $B$ and vice-versa.
i.e. $$n(A) = n(B)$$
Here, it is not necessary that both sets contain same elements.
Note: Equal sets are equivalent but equivalent sets need not be equal.
Illustration 10. Which of the following are equal sets. Justify the answer.
a. Set of letters forming the word DECIMAL and MEDICAL
b. $A = \{ a,e,i,o,u\} $ and $B = \{ x:x\;is\;a\;vowel\;in\;STATUE\} $
c. $A = \{ x:\;x\;is\;a\;letter\;in\;the\;word\;RELATION\} $ and $B = \{ x:x\;is\;a\;letter\;in\;the\;word\;DISTANCE\} $
d. Set of natural numbers less than $10$ and set of single digit natural numbers
e. Set of all two digit numbers ending with $0$ and set of numbers that are multiple of $9$
f. Set of primary colors and set of colors in the flag of France
Solution:
| SET | CLASSIFICATION |
| a. Set of letters forming the word DECIMAL and MEDICAL | Elements of the set $A$ forming the word $DECIMAL$ is $A,C,D,E,I,L,M$ The number of elements in this set $A$ is $7$. Elements of the set $B$ forming the word $MEDICAL$ is $A,C,D,E,I,L,M$ The number of elements in this set $B$ is $7$. $n(A) = n(B)$. $\forall \;x \in A,\;x \in B$ and $\forall \;x \in B,\;x \in A$ Hence, they are equal sets, i.e. $A = B$ |
| b. $A = \{ a,e,i,o,u\} $ and $B = \{ x:x\;is\;a\;vowel\;in\;STATUE\} $ | Elements of the set $A$ is $A,E,I,O,U$ The number of elements in this set $A$ is $5$. Elements of the set $B$ is $A,E,U$ The number of elements in this set $B$ is $3$. $n(A) \ne n(B)$. $\forall \;x \in A,\;x \notin B$ Hence, they are not equal sets, i.e. $A \ne B$ |
| c. $A = \{ x:\;x\;is\;a\;letter\;in\;the\;word\;RELATION\} $ and $B = \{ x:x\;is\;a\;letter\;in\;the\;word\;DISTANCE\} $ | Elements of the set $A$ forming the word $RELATION$ is $A,E,I,L,N,O,R,T$ The number of elements in this set $A$ is $8$. Elements of the set $B$ forming the word $DISTANCE$ is $A,C,D,E,I,N,S,T$ The number of elements in this set $B$ is $8$. $n(A) = n(B)$. $\forall \;x \in A,\;x \notin B$ and $\forall \;x \in B,\;x \notin A$ Hence, they are not equal sets, i.e. $A \ne B$ |
| d. Set of natural numbers less than $10$ and set of single digit natural numbers | Elements of the set $A$ is $1,2,3,4,5,6,7,8,9$ The number of elements in this set $A$ is $9$. Elements of the set $B$ is $1,2,3,4,5,6,7,8,9$ The number of elements in this set $B$ is $9$. $n(A) = n(B)$. $\forall \;x \in A,\;x \in B$ and $\forall \;x \in B,\;x \in A$ Hence, they are equal sets, i.e. $A = B$ |
| e. Set of all two digit numbers ending with $0$ and set of numbers that are multiple of $9$ less than $100$ | Elements of the set $A$ is, $10,20,30,40,50,60,70,80,90 $ The number of elements in this set $A$ is $9$. Elements of the set $B$ is, $18,27,36,45,54,63,72,81$ The number of elements in this set $B$ is $8$. $n(A) \ne n(B)$. $\forall \;x \in A,\;x \notin B$ and $\forall \;x \in B,\;x \notin A$ Hence, they are not equal sets, i.e. $A \ne B$ |
| f. Set of primary colors and set of colors in the flag of France | Elements of the set $A$ is, Red,Blue,Yellow The number of elements in this set $A$ is $3$. Elements of the set $B$ is, Red, Blue, White The number of elements in this set B is $3$. $n(A) = n(B)$. $\forall \;x \in A,\;x \notin B$ and $\forall \;x \in B,\;x \notin A$ Hence, they are not equal sets. |
Illustration 11. Which of the following are equivalent sets. Justify the answer.
a. $A = \{ x:\;x\;is\;a\;letter\;in\;the\;word\;RELATION\} $ and $B = \{ x:x\;is\;a\;letter\;in\;the\;word\;DISTANCE\} $
b. $A = \{ x:\;x < 10,\,x \in N\} $ and $B = \{ x:x < 10,\;x \in Z\} $
c. Set of letters forming the word DECIMAL and MEDICAL
d. Set of primary colors and set of colors in the flag of France
e. $A = \{ x:\;x\,a\,letter\;in\;the\;word\;STEEL\} $ and $B = \{ x:\;x\,a\,letter\;in\;the\;word\;STEAL\} $
Solution:
| SET | CLASSIFICATION |
| a. $A = \{ x:\;x\;is\;a\;letter\;in\;the\;word\;RELATION\} $ and $B = \{ x:x\;is\;a\;letter\;in\;the\;word\;DISTANCE\} $ | Elements of the set $A$ forming the word $RELATION$ is $A,E,I,L,N,O,R,T$ The number of elements in this set $A$ is $8$. Elements of the set $B$ forming the word $DISTANCE$ is $A,C,D,E,I,N,S,T$ The number of elements in this set $B$ is $8$. $n(A) = n(B)$. Hence, they are equivalent sets. |
| b. $A = \{ x:\;x < 10,\,x \in N\} $ and $B = \{ x:x < 10,\;x \in W\} $ | Elements of the set $A$ is $1,2,3,4,5,6,7,8,9$ The number of elements in this set $A$ is $9$. Elements of the set $B$ is $0,1,2,3,4,5,6,7,8,9$ The number of elements in this set $B$ is $10$. $n(A) \ne n(B)$. Hence, they are not equivalent sets. |
| c. Set of letters forming the word DECIMAL and MEDICAL | Elements of the set $A$ forming the word $DECIMAL$ is $A,C,D,E,I,L,M$ The number of elements in this set $A$ is $7$. Elements of the set $B$ forming the word $MEDICAL$ is $A,C,D,E,I,L,M$ The number of elements in this set $B$ is $7$. $n(A) = n(B)$. Hence, they are equivalent sets. |
| d. Set of primary colors and set of colors in the flag of France | Elements of the set $A$ is, Red,Blue,Yellow. The number of elements in this set $A$ is $3$. Elements of the set $B$ is, Red, Blue, White The number of elements in this set B is $3$. $n(A) = n(B)$. Hence, they are equivalent sets. |
| e. $A = \{ x:\;x\,a\,letter\;in\;the\;word\;STEEL\} $ and $B = \{ x:\;x\,a\,letter\;in\;the\;word\;STEAL\} $ | Elements of the set $A$ forming the word $STEEL$ is $E,L,S,T$ The number of elements in this set $A$ is $4$. Elements of the set $B$ forming the word $STEAL$ is $A,E,LS,T$ The number of elements in this set $B$ is $5$. $n(A) \ne n(B)$. Hence, they are not equivalent sets. |
Cardinal number
The number of elements $n$ in a finite set is called the cardinal number of the set and is denoted as $$n(A)$$
Illustration 12. Define the cardinal number for each set.
a. $\{ x:x < 7,\,x \in N\} $
b. Set of consonants in the word MYRMIDON
c. Set of letters in the word RECTIFICATION
d. $A = \{ x: - 10 < x < 10\} $
e. $A = \{ x:x < 200,x \in N\} $
f. Set of cricket players in a team
Solution:
| SET | CARDINAL NUMBER |
| a. $A = \{ x:x < 7,\,x \in N\} $ | The elements of the set are $1,2,3,4,5,6$. Hence, $n(A) = 6$. |
| b. $B$ = Set of consonants in the word MYRMIDON | The consonants in the word $MYRMIDON$ are $D,M,N,R,Y$. Hence, $n(B) = 5$. |
| c. $C$ = Set of letters in the word RECTIFICATION | The letters of the word $RECTIFICATION$ are $A,C,E,F,I,N,O,R,T$ Hence, $n(C) = 9$. |
| d. $D = \{ x: - 10 < x < 10\} $ | The elements of the set are $-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9$ Hence, $n(D) = 19$. |
| e. $E = \{ x:199<x < 200,x \in N\} $ | The given set is a null set. The number of elements is, $n(E) = 0$. |
| f. $F$ = Set of cricket players in a team | There are $11$ players in any cricket team. Hence, $n(F) = 11$. |
