Class 11 | CBSE | MATHEMATICS
COMPLETE SOLUTIONS

Question

Which of the following are examples of the null set (∅)?

1. Set of odd natural numbers divisible by 2
2. Set of even prime numbers
3. {x : x is a natural number, x < 5 and x > 7}
4. {y : y is a point common to any two parallel lines}

Solution (Step-by-Step)

(i) Set of odd natural numbers divisible by 2

Step 1: Odd natural numbers are

\[ 1, 3, 5, 7, \ldots \]

Step 2: A number divisible by 2 must be even.

Conclusion: No odd natural number is divisible by 2.

\[ \Rightarrow \text{This set has no elements} \]

Result: ✅ Null set (∅)

(ii) Set of even prime numbers

Step 1: Prime numbers have exactly two factors.

Step 2: The only even number that is prime is:

\[ 2 \]

Conclusion: The set contains one element.

\[ \{2\} \]

Result: ❌ Not a null set

(iii) \( \{ x : x \in \mathbb{N},\; x < 5 \text{ and } x > 7 \} \)

Step 1: Numbers less than 5 are:

\[ 1, 2, 3, 4 \]

Step 2: Numbers greater than 7 are:

\[ 8, 9, 10, \ldots \]

Step 3: No natural number can satisfy both conditions at the same time.

\[ x < 5 \;\text{and}\; x > 7 \quad \text{(Impossible)} \]

Result: ✅ Null set (∅)

(iv) { y : y is a point common to any two parallel lines}

Step 1: Parallel lines never intersect.

Step 2: Since they do not intersect, they have:

\[ \text{No common point} \]

Result: ✅ Null set (∅)

✅ Final Answer

The null sets are:

\[ \boxed{(i),\ (iii),\ (iv)} \]

Question

Which of the following sets are finite or infinite?

1. The set of months of a year
2. \(\{1, 2, 3, \ldots\}\)
3. \(\{1, 2, 3, \ldots, 99, 100\}\)
4. The set of positive integers greater than 100
5. The set of prime numbers less than 99

Solution (With Reasons)

(i) The set of months of a year

Step 1: Months in a year are:

January, February, March, April, May, June, July, August, September, October, November, December

Step 2: The total number of months is:

\[ 12 \]

Conclusion: The number of elements is fixed and countable.

Result: ✅ Finite set

(ii) \(\{1, 2, 3, \ldots\}\)

Step 1: This set represents all natural numbers.

Step 2: The dots \((\ldots)\) indicate that numbers continue forever.

Conclusion: There is no last element in this set.

Result: ❌ Infinite set

(iii) \(\{1, 2, 3, \ldots, 99, 100\}\)

Step 1: The set starts at 1 and ends at 100.

Step 2: Total number of elements is:

\[ 100 \]

Conclusion: The set has a fixed and definite number of elements.

Result: ✅ Finite set

(iv) The set of positive integers greater than 100

Step 1: Positive integers greater than 100 are:

\[ 101, 102, 103, \ldots \]

Step 2: There is no greatest positive integer.

Conclusion: The elements continue endlessly.

Result: ❌ Infinite set

(v) The set of prime numbers less than 99

Step 1: Prime numbers less than 99 are:

\[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, \] \[ 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 \]

Step 2: The list ends at 97 (the largest prime less than 99).

Conclusion: The number of prime numbers here is limited.

Result: ✅ Finite set

✅ Final Classification

Finite sets:

\[ \boxed{(i),\ (iii),\ (v)} \]

Infinite sets:

\[ \boxed {(ii),\ (iv)} \]

Question 3

State whether each of the following sets is finite or infinite. Give proper reasons.

1. The set of lines which are parallel to the x-axis
2. The set of letters in the English alphabet
3. The set of numbers which are multiples of 5
4. The set of animals living on the earth
5. The set of circles passing through the origin \((0,0)\)

Solution (With Reasons)

(i) The set of lines which are parallel to the x-axis

Step 1: The general equation of a line parallel to the x-axis is:

\[ y = c \]

where \(c\) is any real number.

Step 2: Since \(c\) can take infinitely many real values, there can be infinitely many such lines.

Conclusion: The number of lines parallel to the x-axis is not fixed.

Result: ❌ Infinite set

(ii) The set of letters in the English alphabet

Step 1: The English alphabet consists of:

\[ 26 \]

letters.

Step 2: The number of letters is fixed and definite.

Conclusion: The set has a limited number of elements.

Result: ✅ Finite set

(iii) The set of numbers which are multiples of 5

Step 1: Multiples of 5 are:

\[ 5, 10, 15, 20, 25, \ldots \]

Step 2: There is no greatest multiple of 5.

Conclusion: The numbers continue endlessly.

Result: ❌ Infinite set

(iv) The set of animals living on the earth

Step 1: At any given time, the number of animals living on earth is large but limited.

Step 2: No new animals can exist beyond the actual living population at a time.

Conclusion: The number of elements is definite.

Result: ✅ Finite set

(v) The set of circles passing through the origin \((0,0)\)

Step 1: The general equation of a circle passing through the origin can be written as:

\[ x^2 + y^2 + 2gx + 2fy = 0 \]

where \(g\) and \(f\) are real numbers.

Step 2: Since \(g\) and \(f\) can take infinitely many values, infinitely many circles can pass through \((0,0)\).

Conclusion: The set does not have a fixed number of elements.

Result: ❌ Infinite set

✅ Final Answer Summary

Finite sets:

\[ (ii),\ (iv) \]

Infinite sets:

\[ (i),\ (iii),\ (v) \]

Question 4

In the following, state whether A = B or not. Give proper reasons.

1. \( A = \{ a, b, c, d \} \), \( B = \{ d, c, b, a \} \)
2. \( A = \{ 4, 8, 12, 16 \} \), \( B = \{ 8, 4, 16, 18 \} \)
3. \( A = \{ 2, 4, 6, 8, 10 \} \), B = { x : x is a positive even integer and x ≤ 10 }
4. \( A = \{ x : x \text{ is a multiple of } 10 \} \), \( B = \{ 10, 15, 20, 25, 30, \ldots \} \)

Solution (With Reasons)

(i)

Given:

\[ A = \{ a, b, c, d \} \]

\[ B = \{ d, c, b, a \} \]

Step 1: Check the elements of both sets.

Both A and B contain exactly the same elements.

Step 2: Order of elements does not matter in a set.

Conclusion: \( A = B \)

Result: ✅ A = B

(ii)

Given:

\[ A = \{ 4, 8, 12, 16 \} \]

\[ B = \{ 8, 4, 16, 18 \} \]

Step 1: Compare the elements.

Element \(12\) is in set A but not in set B.

Element \(18\) is in set B but not in set A.

Conclusion: Both sets do not contain the same elements.

Result: ❌ A ≠ B

(iii)

Given:

\[ A = \{ 2, 4, 6, 8, 10 \} \]

B ={ x : x is a positive even integer and x ≤ 10 }

Step 1: Write B in roster form.

\[ B = \{ 2, 4, 6, 8, 10 \} \]

Step 2: Compare A and B.

Both sets contain exactly the same elements.

Conclusion: \( A = B \)

Result: ✅ A = B

(iv)

Given:

\[ A = \{ x : x \text{ is a multiple of } 10 \} \]

\[ B = \{ 10, 15, 20, 25, 30, \ldots \} \]

Step 1: Write A in roster form.

\[ A = \{ 10, 20, 30, 40, \ldots \} \]

Step 2: Observe set B.

B contains numbers like \(15\) and \(25\) which are not multiples of 10.

Conclusion: A and B do not have the same elements.

Result: ❌ A ≠ B

✅ Final Summary

\[ (i)\ A = B,\quad (ii)\ A \ne B,\quad (iii)\ A = B,\quad (iv)\ A \ne B \]

Question 5

Are the following pair of sets equal? Give reasons.

1. \( A = \{2,\;3\} \)
   \( B = \{x : x \text{ is a solution of } x^2 + 5x + 6 = 0\} \)

2. \( A = \{ x : x \text{ is a letter in the word FOLLOW} \} \)
   \( B = \{ y : y \text{ is a letter in the word WOLF} \} \)

Solution (Step-by-Step)

✅ Final Answer

\[ \boxed{(i)\ A \ne B} \]

\[ \boxed{(ii)\ A = B} \]

Question 6

From the sets given below, select the equal sets.

\( A = \{2,\;4,\;8,\;12\} \)
\( B = \{1,\;2,\;3,\;4\} \)
\( C = \{4,\;8,\;12,\;14\} \)
\( D = \{3,\;1,\;4,\;2\} \)

\( E = \{-1,\;1\} \)
\( F = \{0,\;a\} \)
\( G = \{1,\;-1\} \)
\( H = \{0,\;1\} \)

✏️ Solution (Step-by-Step)

✅ Final Answer

Equal sets are:

\[ B = D \]

\[ E = G \]