Set Examples

1) Which of the following sets is(are) infinite?

• Set of all Indian Nobel laureates
• Set of squares of all odd natural numbers
• Set of all countries in the world
• Set of all leap years

Detailed Answer:

• Set of Indian Nobel laureates: Finite (limited individuals).
• Set of squares of all odd natural numbers: Infinite (there are infinitely many odd numbers).
• Set of all countries in the world: Finite.
• Set of all leap years: Infinite (calendar continues indefinitely).

Correct Answers:

• Set of squares of all odd natural numbers
• Set of all leap years

2) Which of the following set comprehension defines real numbers in interval $[-2,0) \cup (4,8]$?

A: $ {r \mid r \in \mathbb{R}, -2 \le r < 8} $ B: $ {r \mid r \in \mathbb{R}, -2 \le r < 0 and 4 < r \le 8} $ C: $ {r \mid r \in \mathbb{R}, -2 \le r < 0 or 4 < r \le 8} $ D: $ {r \mid r \in \mathbb{R}, -2 \le r < 0 and 4 \le r \le 8} $

Detailed Answer:

We require $r$ to be in $[-2,0)$ or in $(4,8]$. This means “or,” not “and”.

Correct Answer: C

3) Which of the following set comprehensions define squares of the first 100 natural numbers?

• $ { n \mid n \in \mathbb{N}, \sqrt{n} \in \mathbb{N} and n < 100 } $
• $ { n^2 \mid n \in \mathbb{N}, n < 100 } $
• $ { n \mid n \in \mathbb{N}, \sqrt{n} \in \mathbb{N} and n < 10000 } $
• $ { n^2 \mid n \in \mathbb{N}, n < 10000 } $

Detailed Answer:

The set of all numbers that can be written as $ n^2 $, with $ n $ a natural number less than 100:

This gives us all squares from $1^2 = 1$ to $99^2 = 9801$, a total of 99 numbers.

Correct Answer: Second option

4) Which of the following statement(s) is(are) true?

• Empty set contains only one element i.e. $ \phi $.
• Set with $ n $ elements has $2^n-1$ subsets.
• Empty set is an element in the power set of a set.
• Power set is a set of all the subsets of a set.

Detailed Answer:

• Empty set contains no elements (so the first is FALSE).
• Power set contains $2^n$ subsets (not $2^n-1$), so this is FALSE.
• The empty set is always an element of the power set (TRUE).
• The power set is indeed the set of all subsets (TRUE).

Correct Answers:

• Empty set is an element in the power set of a set.
• Power set is a set of all the subsets of a set.

5) Which of the following intervals are subsets of ?[^1]

• $((0, 2.3] \cup (2.3, 3]) \cap (0, 2.3))$
• $((2, 2.5] \cup (2.5, 4]) \setminus (0, 2.3))$
• $((2, 2.5] \cup (2.5, 4]) \cap (0, 3))$
• $((0, 2.3] \cup (2.2, 3)) \setminus (0, 2.3))$

Detailed Answer:

Looking for intervals entirely contained in.[^1]

• (2, 2.5] is within ; (2.5, 4] is not because 4 > 3.[^1]
• The only part within is (2, 2.5].[^1]
• Option 3 only keeps pieces within (0, 3): (2, 2.5] is kept; (2.5, 4] gets cut to (2.5, 3).

Thus, interval $((2, 2.5] \cup (2.5, 4]) \cap (0, 3)) = (2, 2.5] \cup (2.5, 3)$ is a subset of.[^1]

Correct Answer: Third option

6) Let $ M $ be the set of all real numbers that are strictly greater than 6 or strictly less than -6. How can we represent $ B $ in set comprehension form, where $ B $ is a subset of $ M $ and has only integers as elements?

• $ B = { z \mid z \in \mathbb{Z}, z \in [-6, 6]} $
• $ B = { z \mid z \in \mathbb{Z}, z \in (-6, 6)} $
• $ B = { z \mid z \in \mathbb{Z}, z \in (-\infty, -6) \cup (6, \infty)} $
• $ B = {z \mid z \in \mathbb{Z}, z \in (-\infty, -6] \cup [6, \infty)}$

Detailed Answer:

$ M = {x \in \mathbb{R} \mid x < -6 or x > 6} $. $ B $ is the subset consisting only of integers.

So, $ B = { z \mid z \in \mathbb{Z}, z < -6 or z > 6} = { z \mid z \in \mathbb{Z}, z \in (-\infty, -6) \cup (6, \infty)} $.

Correct Answer: Third option.

7) Two finite sets $ A $ and $ B $ are such that the total number of subsets of $ A $ is 56 more than that of $ B $. What are the cardinalities of $ A $ and $ B $ respectively?

Options:

• 6, 3
• 8, 4
• 7, 1
• 8, 3

Detailed Answer:

Let $ |A| = m, |B| = n $.

Number of subsets: $ 2^m = 2^n + 56 $

Try possible combinations:

$ n = 3: 2^3 = 8 $ $ m = 6: 2^6 = 64; 64 - 8 = 56 $

So, $ |A| = 6, |B| = 3 $.

Correct Answer: 6, 3