**Topic: Sets and Their Types**

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

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

(ii)

*B*={

*is a letter in the word ”MARS”}*

*x∶x*(iii)

*C*={

*y∶y*is a letter in the word ”ARMS’”}

(iv)

*D*={

*x*∶3

*x*-2=0,

*x*∈ℚ}

(v)

*E*={

*∈ℕ and*

*x∶x**x*is an odd number}

✍ Solution:

(i)

*A*={

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

*x∶x*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 “MARS”}*

*x∶x*⇒

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

(iii)

*C*={

*y∶y*is a letter in the word “ARMS”}

⇒

*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*∈ℚ}

⇒

*D*={⅔}

*x*-2=0⇒

*x*=⅔∈ℚ]

Hence,

*D*is a singleton set.

(v)

*E*={

*∈ℕ and*

*x∶x**x*is an odd number}.

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

**Types of Sets**

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

(i) *A*={* x∶x*<1 and

*x*>3}

(ii)

*B*={

*x∶x*^{3}-1=0,

*x*∈ℝ}

(iii)

*C*={

*x∶x*∈ℕ and

*x*is a prime number}

(iv)

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

(v)

*E*={

*is positive even integer and*

*x∶x**x*≤10}

✍ Answers:

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

📌 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.

✍

*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*⊆*B* nor *B*⊆*A*.

(v) {Letters of English alphabet}.

👑 Particular Symbols ℕ 𝕎 ℤ ℚ 𝕋 ℝ ℂ Represent Number Sets in Mathematics 💎