1. Sets

1. Decide which of the following are equal sets and which are not ? Justify your answer.
A = { x | 3x − 1 = 2}
B = { x | x is a natural number but x is neither prime nor composite.}
C = { x | x ∈ N, x < 2}

Solution:

First, let’s write the given sets in the list form:

A = { x | 3x − 1 = 2}
3x − 1 = 2
∴ 3x = 2 + 1
∴ 3x = 3
∴ x = \(\displaystyle \frac{3}{3}\)
∴ x = 1
∴ A = {1} ... (i)

And B = { x | x is a natural number but x is neither prime nor composite.}
∴ x = 1
∴ B = {1} ... (ii)

And C = { x | x ∈ N, x < 2}
∴ x = 1
∴ C = {1} ... (iii)

From (i), (ii), and (iii), we have:
Sets A, B, and C have exactly the same element(s).
∴ A = B = C
∴ Sets A, B, and C are equal sets.

2. Decide whether set A and B are equal sets. Give reason for your answer.
A = Even prime numbers
B = {x | 7x − 1 = 13}

Solution:

First, let's write the given sets in the list form:

A = {2} ... (Since 2 is the only even prime number) ... (i)
B = {x | 7x − 1 = 13}
7x − 1 = 13
∴ 7x = 13 + 1
∴ 7x = 14
∴ x = \(\displaystyle \frac{14}{7}\)
∴ x = 2
∴ B = {2} ... (ii)

From (i) and (ii),
Sets A and B have exactly the same element(s).
∴ A = B
∴ Sets A and B are equal sets.

3. Which of the following are empty sets? Why?

(i) A = { a | a is a natural number smaller than 0}

Solution:

There is no natural number which is less than 0.
∴ A is an empty set.

(ii) B = { x | x² = 0 }

Solution:

B = {0}
Thus, B is a set which contains an element 0.
∴ B is not an empty set.

(iii) C = { x | 5x − 2 = 0, x ∈ N }

Solution:

5x − 2 = 0

∴ 5x = 2

∴ x = \(\displaystyle \frac{2}{5}\)

But, x ∈ N

and \(\displaystyle \frac{2}{5}\) is not a natural number.

∴ C is an empty set.

4. Write with reasons, which of the following sets are finite or infinite:

(i) A = { x | x < 10, x is a natural number }

Solution:

A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
∴ A is a set which contains 10 elements.
∴ A is a finite set.

(ii) B = { y | y < − 1, y is an integer }

Solution:

B = { − 2, − 3, − 4, − 5, ... }
Thus, B is a set which contains infinitely many elements.
∴ B is an infinite set.

(iii) C = Set of students of class 9 from your school.

Solution:

C = {students of class 9 from your school}
Thus, C is a set which contains a definite number of elements.
∴ C is a finite set.

(iv) Set of people from your village.

Solution:

This is a set which contains a definite number of elements.
∴ This is a finite set.

(v) Set of apparatus in laboratory

Solution:

This is a set which contains a definite number of elements.
∴ This is a finite set.

(vi) Set of whole numbers

Solution:

W = { 0, 1, 2, 3, ... }
Thus, W is a set which contains infinitely many elements.
∴ W is an infinite set.

(vii) Set of rational numbers

Solution:

This is an infinite set.

1. Decide which of the following are equal sets and which are not ? Justify your answer. A = { x | 3x − 1 = 2} B = { x | x is a natural number but x is neither prime nor composite.} C = { x | x ∈ N, x < 2}

Solution:

2. Decide whether set A and B are equal sets. Give reason for your answer. A = Even prime numbers B = {x | 7x − 1 = 13}

Solution:

3. Which of the following are empty sets? Why?

(i) A = { a | a is a natural number smaller than 0}

Solution:

(ii) B = { x | x² = 0 }

Solution:

(iii) C = { x | 5x − 2 = 0, x ∈ N }

Solution:

4. Write with reasons, which of the following sets are finite or infinite:

(i) A = { x | x < 10, x is a natural number }

Solution:

(ii) B = { y | y < − 1, y is an integer }

Solution:

(iii) C = Set of students of class 9 from your school.

Solution:

(iv) Set of people from your village.

Solution:

(v) Set of apparatus in laboratory

Solution:

(vi) Set of whole numbers

Solution:

(vii) Set of rational numbers

Solution:

1. Decide which of the following are equal sets and which are not ? Justify your answer.
A = { x | 3x − 1 = 2}
B = { x | x is a natural number but x is neither prime nor composite.}
C = { x | x ∈ N, x < 2}

2. Decide whether set A and B are equal sets. Give reason for your answer.
A = Even prime numbers
B = {x | 7x − 1 = 13}