Maharashtra Board Class 9 Maths Chapter 1 Set 1.2 Algebra Standard Part 1 Sets Solutions

Question 1. Decide which of the following are equal sets and which are not? Justify your answer.
\( A = \{x \mid 3x - 1 = 2\} \)
\( B = \{x \mid x \text{ is a natural number but } x \text{ is neither prime nor composite}\} \)
\( C = \{x \mid x \in \mathbb{N}, x < 2\} \)
Answer:
For set \( A \):
\( 3x - 1 = 2 \)

\( \implies 3x = 2 + 1 \)

\( \implies 3x = 3 \)

\( \implies x = 1 \)
Therefore, \( A = \{1\} \) — (i)

For set \( B \):
\( x \) is a natural number which is neither prime nor composite.
The only natural number that is neither prime nor composite is \( 1 \).
Therefore, \( B = \{1\} \) — (ii)

For set \( C \):
\( x \in \mathbb{N} \) and \( x < 2 \).
The only natural number less than \( 2 \) is \( 1 \).
Therefore, \( C = \{1\} \) — (iii)

From (i), (ii), and (iii), the elements of sets \( A \), \( B \), and \( C \) are identical.
Hence, \( A = B = C \), meaning they are equal sets.
In simple words: Equal sets are sets that contain the exact same elements. Since solving each condition gives us the number 1 as the only member, all three sets are equal.

🎯 Exam Tip: To score full marks, always write down the roster form of each set clearly and label them as equations (i), (ii), and (iii) before making your final comparison.

Question 1. Decide whether the following sets are equal sets or not. Give reason for 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}
Answer:
\( A = \{x \mid 3x - 1 = 2\} \)
Here, \( 3x - 1 = 2 \)
\( \implies 3x = 3 \)
\( \implies x = 1 \)
\( \therefore A = \{1\} \) ... (i)

\( B = \{x \mid x \text{ is a natural number but } x \text{ is neither prime nor composite}\} \)
Since 1 is the only natural number which is neither prime nor composite,
\( \implies x = 1 \)
\( \therefore B = \{1\} \) ... (ii)

\( C = \{x \mid x \in N, x < 2\} \)
Since 1 is the only natural number less than 2,
\( \implies x = 1 \)
\( \therefore C = \{1\} \) ... (iii)

From (i), (ii), and (iii), the elements in sets A, B, and C are identical.
\( \therefore \) A, B, and C are equal sets.
In simple words: Equal sets are sets that contain the exact same elements. Since solving each set gives us the single number 1, all three sets are equal.

🎯 Exam Tip: Clearly label your simplified sets as (i), (ii), and (iii) before concluding they are equal. This structured approach helps the examiner award full marks easily.

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:
\( A = \text{Even prime numbers} \)
Since 2 is the only even prime number,
\( \therefore A = \{2\} \) ... (i)

\( B = \{x \mid 7x - 1 = 13\} \)
Here, \( 7x - 1 = 13 \)
\( \implies 7x = 14 \)
\( \implies x = 2 \)
\( \therefore B = \{2\} \) ... (ii)

From (i) and (ii), the element in set A and B is identical.
\( \therefore \) A and B are equal sets.
In simple words: Set A contains only the number 2 because it is the only even prime number. Solving the equation for Set B also gives us 2. Since both sets contain the exact same element, they are equal.

🎯 Exam Tip: Remember that 2 is the only even prime number in existence. Keep this fact in mind as it is frequently tested in set theory questions.

Question 3. Which of the following are empty sets? Why?
i. A = {a | a is a natural number smaller than zero}
ii. B = {x | x² = 0}
Answer:
i. \( A = \{a \mid a \text{ is a natural number smaller than zero}\} \)
Natural numbers start from 1 (i.e., 1, 2, 3, ...). There is no natural number smaller than zero.
\( \therefore A = \{ \} \) (or \( \emptyset \))
Thus, set A is an empty set.

ii. \( B = \{x \mid x^2 = 0\} \)
Here, \( x^2 = 0 \)
\( \implies x = 0 \)
\( \dots B = \{0\} \)
Since set B contains the element 0, it is not empty.
Thus, set B is not an empty set.
In simple words: An empty set has absolutely no elements inside it. Set A is empty because no natural number is less than zero, but Set B is not empty because it contains the number 0.

🎯 Exam Tip: Do not confuse the set \( \{0\} \) with an empty set \( \{ \} \). A set containing zero has one element, so it is a singleton set, not an empty set.

Question 3. State which of the following sets are empty sets and why.
(i) A = {a | a is a natural number smaller than zero}
(ii) B = {x | x² = 0}
(iii) C = {x | 5x - 2 = 0, x ∈ N}
Answer:
(i) A = {a | a is a natural number smaller than zero}
Natural numbers begin from 1.
\( \implies \) A = { }
\( \implies \) A is an empty set.

(ii) B = {x | x² = 0}
Here, \( x^2 = 0 \)
\( \implies \) x = 0 [Taking square root on both sides]
\( \implies \) B = {0}
\( \implies \) B is not an empty set because it contains exactly one element, which is zero.

(iii) C = {x | 5x - 2 = 0, x ∈ N}
Here, \( 5x - 2 = 0 \)
\( \implies 5x = 2 \)
\( \implies x = \frac{2}{5} \)
Given, \( x \in \mathbb{N} \)
But, \( x = \frac{2}{5} \) is not a natural number.
\( \implies \) C = { }
\( \implies \) C is an empty set.
In simple words: An empty set is a set that has absolutely no elements in it. Since there are no natural numbers less than zero, and no natural number can satisfy the equation in set C, both A and C are empty sets, while B contains the number 0.

🎯 Exam Tip: Always write the elements of the set first to clearly show whether it contains any members or is empty.

Question 4. Write with reasons, which of the following sets are finite or infinite.
(i) A = {x | x < 10, x is a natural number}
(ii) B = {y | y < -1, y is an integer}
(iii) C = Set of students of class 9 from your school.
(iv) Set of people from your village.
(v) Set of apparatus in laboratory
(vi) Set of whole numbers
(vii) Set of rational numbers
Answer:
(i) A = {x | x < 10, x is a natural number}
\( \implies \) A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
The number of elements in A is limited and can be counted.
\( \implies \) A is a finite set.

(ii) B = {y | y < -1, y is an integer}
\( \implies \) B = {..., -4, -3, -2}
The number of elements in B is unlimited and cannot be counted.
\( \implies \) B is an infinite set.

(iii) C = Set of students of class 9 from your school.
The number of students in class 9 of any school is limited and can be counted.
\( \implies \) C is a finite set.

(iv) Set of people from your village.
The population of a village is limited and can be counted, even if the number is very large.
\( \implies \) This is a finite set.

(v) Set of apparatus in laboratory
The number of apparatus in a laboratory is limited and can be counted.
\( \implies \) This is a finite set.

(vi) Set of whole numbers
Whole numbers are W = {0, 1, 2, 3, ...}.
The number of elements is unlimited and cannot be counted.
\( \implies \) This is an infinite set.

(vii) Set of rational numbers
The number of rational numbers is unlimited and cannot be counted.
\( \implies \) This is an infinite set.
In simple words: A set is finite if we can count all its elements and the counting stops. If the elements go on forever without end, like whole numbers or integers, the set is infinite.

🎯 Exam Tip: To determine if a set is finite or infinite, try listing its elements; if the list ends, it is finite, and if it has three dots (...) at the end with no closing limit, it is infinite.

Question. Classify the following sets as finite or infinite:
(ii) \( B = \{y \mid y < -1, y \text{ is an integer}\} \)
(iii) C = Set of students of class 9 from your school.
(iv) Set of people from your village.
(v) Set of apparatus in laboratory.
(vi) Set of whole numbers.
(vii) Set of rational numbers.
Answer:
(ii) \( B = \{y \mid y < -1, y \text{ is an integer}\} \)
\( \implies B = \{\dots, -4, -3, -2\} \)
The number of elements in B is unlimited and uncountable.
\( \implies \) B is an infinite set.

(iii) C = Set of students of class 9 from your school.
The number of students in a class is limited and can be counted.
\( \implies \) C is a finite set.

(iv) Set of people from your village.
The number of people in a village is limited and can be counted.
\( \implies \) Given set is a finite set.

(v) Set of apparatus in laboratory.
The number of apparatus in the laboratory are limited and can be counted.
\( \implies \) Given set is a finite set.

(vi) Set of whole numbers.
The number of elements in the set of whole numbers are unlimited and uncountable.
\( \implies \) Given set is an infinite set.

(vii) Set of rational numbers.
The number of elements in the set of rational numbers are unlimited and uncountable.
\( \implies \) Given set is an infinite set.
In simple words: A set is finite if we can count all its members, and infinite if the counting never ends because there are too many members.

🎯 Exam Tip: Remember that sets of standard numbers like whole numbers, natural numbers, integers, and rational numbers are always infinite because they go on forever.

Question 1. If \( A = \{1, 2, 3\} \) and \( B = \{1, 2, 3, 4\} \), then \( A \neq B \) verify it. (Textbook pg. no. 6)
Answer: Here, \( 4 \in B \) but \( 4 \notin A \). This single difference shows that the two sets do not contain the exact same elements.
\( \implies A \) and \( B \) are not equal sets, i.e. \( A \neq B \).
In simple words: Two sets are equal only if they have the exact same elements. Since set B has the number 4 but set A does not, they are not equal.

🎯 Exam Tip: To prove two sets are unequal, it is enough to find just one element that belongs to one set but not the other.

Question 2. \( A = \{x \mid x \text{ is a prime number and } 10 < x < 20\} \) and \( B = \{11, 13, 17, 19\} \). Here \( A = B \). Verify. (Textbook pg. no. 6)
Answer: \( A = \{x \mid x \text{ is a prime number and } 10 < x < 20\} \)
\( \implies A = \{11, 13, 17, 19\} \)
Given, \( B = \{11, 13, 17, 19\} \)
Since every element of set A is in set B, and every element of set B is in set A, the sets are completely identical.
\( \implies A = B \)
In simple words: When we list out the prime numbers between 10 and 20, we get 11, 13, 17, and 19. Since this list is exactly the same as set B, both sets are equal.

🎯 Exam Tip: Always convert set-builder form into listing form first to easily compare and verify if two sets are equal.

Question. Decide whether set A and set B are equal sets: \( A = \{x \mid x \text{ is a prime number and } 10 < x < 20\} \) and \( B = \{11, 13, 17, 19\} \).
Answer: \( A = \{x \mid x \text{ is a prime number and } 10 < x < 20\} \)
\( \implies A = \{11, 13, 17, 19\} \)
\( B = \{11, 13, 17, 19\} \)
Since all the elements in set A and B are identical,
\( \implies A \) and \( B \) are equal sets, i.e., \( A = B \).
In simple words: Set A contains prime numbers between 10 and 20, which are 11, 13, 17, and 19. Since Set B has the exact same numbers, both sets are equal.

🎯 Exam Tip: Always list the elements of both sets in roster form first to clearly show the examiner that all elements are identical.