Practice Set 1.2 Sets | Class 9th Mathematics

Practice Set 1.2 Sets Class 9th Mathematics Part I MHB Solution

Practice Set 1.2

Question 1.

Decide which of the following are equal sets and which are not? Justify your answer.

A = \( \{x | 3x – 1 = 2\} \)

B = \( \{x | x \text{ is a natural number but } x \text{ is neither prime nor composite}\} \)

C = \( \{x | x \in N, x < 2\} \)

Answer:

We know that two sets A and B are said to be equal if every element of set A is in set B and every element of set B is in set A. It is symbolically written as \( A = B \).

For Set A: \( A = \{x | 3x – 1 = 2\} \)
Solving the equation: \[ 3x – 1 = 2 \] \[ 3x = 2 + 1 \] \[ 3x = 3 \] \[ x = \frac{3}{3} \] \[ x = 1 \] Therefore, \( A = \{1\} \).

For Set B: \( B = \{x | x \text{ is a natural number but } x \text{ is neither prime nor composite}\} \)
The only natural number that is neither prime nor composite is 1. Therefore, \( B = \{1\} \).

For Set C: \( C = \{x | x \in N, x < 2\} \)
Natural numbers (\(N\)) are typically \( \{1, 2, 3, \dots\} \). However, if we follow the context of some MHB textbooks which might include 0 in N for certain exercises, let's analyze both:

If \(N = \{1, 2, 3, \dots\}\), then natural numbers less than 2 is only 1. So, \( C = \{1\} \).
If \(N = \{0, 1, 2, 3, \dots\}\) (as implied by the original solution's \(C=\{0,1\}\)), then natural numbers less than 2 are 0 and 1. So, \( C = \{0, 1\} \).

Let's proceed with the standard definition \(N = \{1, 2, 3, \dots\}\) as it's more common, and correct the original solution's implication about C. If we strictly follow the original numerical answer for C, the definition of N including 0 must be assumed. Correcting: The original solution states N = {0, 1, 2, ...} and derives C = {0, 1}. Using this assumption for consistency with the original problem's intent for C: The natural numbers \(x \in N\) such that \(x < 2\) are 0 and 1. Therefore, \( C = \{0, 1\} \).

Comparison:
Comparing elements: \( A = \{1\} \)
\( B = \{1\} \)
\( C = \{0, 1\} \) (based on the original solution's assumption for N)

Thus, \( A = B \), but \( A \neq C \) and \( B \neq C \).

Ans. A and B are equal sets. Sets A and C are not equal. Sets B and C are not equal.

Question 2.

Decide whether set A and B are equal sets. Give reason for your answer.

A = Even prime numbers

B = \( \{x | 7x – 1 = 13\} \)

Answer:

Two sets are equal if they have exactly the same elements.

For Set A: A = Even prime numbers.
The only even prime number is 2. Therefore, \( A = \{2\} \).

For Set B: \( B = \{x | 7x – 1 = 13\} \)
Solving the equation: \[ 7x – 1 = 13 \] \[ 7x = 13 + 1 \] \[ 7x = 14 \] \[ x = \frac{14}{7} \] \[ x = 2 \] Therefore, \( B = \{2\} \).

Comparison:
Since \( A = \{2\} \) and \( B = \{2\} \), the sets have the same element.

Ans. Set A and set B are equal sets.

Question 3.

Which of the following are empty sets? Why?

A = \( \{a | a \text{ is a natural number smaller than zero}\} \)
B = \( \{x | x^2 = 0\} \)
C = \( \{x | 5x – 2 = 0, x \in N\} \)

Answer:

An empty set (or null set) is a set that contains no elements. It is denoted by \( \{\} \) or \( \emptyset \).

i. A = \( \{a | a \text{ is a natural number smaller than zero}\} \)

Natural numbers (\(N\)) are typically \( \{1, 2, 3, \dots\} \). Some contexts (like the original source for Q1) use \(N = \{0, 1, 2, \dots\}\). Under the standard definition \(N = \{1, 2, 3, \dots\}\), there are no natural numbers smaller than zero. If \(N = \{0, 1, 2, \dots\}\), there are still no natural numbers strictly smaller than zero. Therefore, there are no elements satisfying this condition. So, \( A = \{\} \).

Ans. A is an empty set.

ii. B = \( \{x | x^2 = 0\} \)

Solving the equation: \[ x^2 = 0 \] \[ x = 0 \] The set B contains the element 0. So, \( B = \{0\} \). Since it contains an element, it is not empty.

Ans. B is not an empty set.

iii. C = \( \{x | 5x – 2 = 0, x \in N\} \)

Solving the equation: \[ 5x – 2 = 0 \] \[ 5x = 2 \] \[ x = \frac{2}{5} \] The value of \(x\) is \( \frac{2}{5} \). However, \(x\) must be a natural number (\(x \in N\)). Since \( \frac{2}{5} \) is not a natural number (whether N starts from 0 or 1), there is no natural number satisfying the condition. So, \( C = \{\} \).

Ans. C is an empty set.

Question 4.

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

A set is finite if it is an empty set or if its elements can be counted and the counting process terminates. Otherwise, the set is infinite.

A = \( \{x | x < 10, x \text{ is a natural number}\} \)
B = \( \{y | y < -1, y \text{ is an integer}\} \)
C = Set of students of class 9 from your school.
Set of people from your village.
Set of apparatus in laboratory.
Set of whole numbers.
Set of rational numbers.

Answer:

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

Assuming Natural numbers \(N = \{1, 2, 3, \dots\}\). Then the natural numbers less than 10 are \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \).

If we assume Natural numbers \(N = \{0, 1, 2, \dots\}\) (as implied by original solution to Q1), then the natural numbers less than 10 are \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \).

In either case, the number of elements is limited and countable (9 or 10 elements). ∴ \( A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \) (using standard N) or \( A = \{0, 1, ..., 9\} \) (using N from original problem's context).

Ans. A is a finite set.

ii. B = \( \{y | y < -1, y \text{ is an integer}\} \)

Integers less than -1 are \( \dots, -5, -4, -3, -2 \). ∴ \( B = \{\dots, -5, -4, -3, -2\} \). The number of elements is unlimited and uncountable in the negative direction.

Ans. B is an infinite set.

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

The number of students in a specific class at a school is countable and limited, even if it's a large number. It cannot be unlimited.

Ans. C is a finite set.

iv. Set of people from your village.

The population of a village is a specific, countable number at any given time. It is limited.

Ans. This set is finite.

v. Set of apparatus in laboratory.

The number of distinct apparatus in a laboratory, while potentially large, is countable and limited.

Ans. This set is finite.

vi. Set of whole numbers.

The set of whole numbers is \( W = \{0, 1, 2, 3, \dots\} \). The number of elements is unlimited and uncountable in the positive direction.

Ans. W is an infinite set.

vii. Set of rational numbers.

A rational number is any number that can be expressed as a fraction \( \frac{p}{q} \) where \(p\) and \(q\) are integers and \( q \neq 0 \). Examples: \( \frac{1}{2}, -3, 0, 0.25 \). There are infinitely many rational numbers (e.g., between any two distinct rational numbers, there is another rational number). The number of elements is unlimited and uncountable.

Ans. The set of rational numbers (Q) is an infinite set.