**Topic: Sets and Their Types**

π Example 1. Identify which of the following set is an **empty set**, **singleton set**, **infinite set** or **equal sets**.

(i) *A*={* xβΆx* is a girl being living on the Jupiter}

(ii)

*B*={

*is a letter in the word βMARSβ}*

*xβΆx*(iii)

*C*={

*yβΆy*is a letter in the word βARMSββ}

(iv)

*D*={

*x*βΆ3

*x*-2=0,

*x*ββ}

(v)

*E*={

*ββ and*

*xβΆx**x*is an odd number}

β Solution:

(i)

*A*={

*is a girl being living on the Jupiter}.*

*xβΆx*We know that, there is no human being or any girl being living on the Jupiter. Hence,

*A*is an empty set.

(ii)

*B*={

*is a letter in the word βMARSβ}*

*xβΆx*β

*B*={M, A, R, S}

(iii)

*C*={

*yβΆy*is a letter in the word βARMSβ}

β

*C*={A, R, M, S}

Here, we observe that the elements of sets

*B*and

*C*are exactly same, hence these sets are equal.

(iv)

*D*={x: 3x-2=0,

*x*ββ}

β

*D*={β }

*x*-2=0β

*x*=β ββ]

Hence,

*D*is a singleton set.

(v)

*E*={

*ββ and*

*xβΆx**x*is an odd number}.

Clearly, it is an infinite set because there are infinite natural numbers which are odd.

**Types of Sets**

π Example 2. From the sets given below, select **empty set**, **singleton set**, **infinite set** and **equal sets**.

(i) *A*={* xβΆx*<1 and

*x*>3}

(ii)

*B*={

*xβΆx*^{3}-1=0,

*x*ββ}

(iii)

*C*={

*xβΆx*ββ and

*x*is a prime number}

(iv)

*D*={2, 4, 6, 8, 10}

(v)

*E*={

*is positive even integer and*

*xβΆx**x*β€10}

β Answers:

(i) Empty set (ii) Singleton set (iii) Infinite set (iv) and (v) Equal sets

π Example 3. Let *A*={letters of BOMBAY} and *B*={letters of CALCUTTA}

(i) Are these sets disjoint or overlapping?

(ii) Are these sets equal?

(iii) Are these sets equivalent?

(iv) Is any of these sets, subset of the other?

(v) Describe a universal set for this problem.

β

*A*={letters of BOMBAY}={A, B, M, O, Y}

*B*={letters of CALCUTTA}={A, C, L, T, U}.

Answers:

(i) These sets are overlapping.

(ii) These sets are not equal.

(iii) These sets are equivalent.

(iv) Neither *A*β*B* nor *B*β*A*.

(v) {Letters of English alphabet}.

π Particular Symbols β π β€ β π β β Represent Number Sets in Mathematics π