Sets and Their Types — Types of Sets

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={x∶x is a letter in the word ”MARS”}
(iii) C={y∶y is a letter in the word ”ARMS’”}
(iv) D={x∶3x-2=0, x∈ℚ}
(v) E={x∶x∈ℕ and x is an odd number}
✍ Solution:
(i) A={x∶x is a girl being living on the Jupiter}.
We know that, there is no human being or any girl being living on the Jupiter. Hence, A is an empty set.
(ii) B={x∶x is a letter in the word “MARS”}
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}

[∵3x-2=0⇒x=⅔∈ℚ]

Hence, D is a singleton set.
(v) E={x∶x∈ℕ and 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∶x3-1=0,x∈ℝ}
(iii) C={x∶x∈ℕ and x is a prime number}
(iv) D={2, 4, 6, 8, 10}
(v) E={x∶x is positive even integer and 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 AB nor BA.
(v) {Letters of English alphabet}.

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

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