NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.2 Sets help students master the classification of sets based on their elements. While Exercise 1.1 focused on representation, this exercise dives deeper into identifying the nature and size of different sets.
Prepared according to the latest CBSE Class 11 Maths syllabus, Exercise 1.2 focuses on crucial concepts like the Empty Set, Finite and Infinite Sets, and Equal Sets. Mastering these is essential for understanding more complex topics like Subsets, Power Sets, and Venn Diagrams.
NCERT Solutions for Class 11 Maths Chapter 1 Sets Exercise 1.2
Q.
Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) {x: x is a natural numbers, x < 5 and x > 7 }
(iv) {y : y is a point common to any two parallel lines}
Q.
Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3, . . .}
(iii) {1, 2, 3, . . .99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
Q.
State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0, 0)
Q.
In the following, state whether A = B or not:
(i) A = {a, b, c, d}; B = { d, c, b, a }
(ii) A = {4, 8, 12, 16}; B = { 8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10}; B = {x: x is positive even integer and x ≤ 10}
(iv) A = { x : x is a multiple of 10};
B = { 10, 15, 20, 25, 30, . . . }
Q.
Examine whether the following statements are true or false:(i) a,b⊂b,c, a(ii) a,e⊂x : x is a vowel in the English alphabet(iii) { 1, 2, 3 }⊂ 1, 3,5 (iv) a⊂a,b, c(v) a∈a,b, c(vi) {x : x is an even natural number less than 6}⊂{x : x is a natural number which divides 36}
Q.
Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A
(ii) {3, 4} ∈ A
(iii) {{3, 4}} ⊂ A
(iv) 1 ∈ A
(v) 1 ⊂ A
(vi) {1, 2, 5} ⊂ A
(vii) {1, 2, 5} ∈ A
(viii) {1, 2, 3} ⊂ A
(ix) Φ ∈ A
(x) Φ ⊂ A
(xi) {Φ} ⊂ A
Q.
Write down all the subsets of the following sets
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) Φ
The solutions are designed to provide a step-by-step logical flow, ensuring that students can distinguish between sets that have no elements and those that have an uncountable number of elements.
Q1. Which of the following are examples of the null set?
- Definition: A set which does not contain any element is called the empty set, null set, or void set.
- (i) Set of odd natural numbers divisible by 2:
- Solution: Since no odd number is divisible by 2, this set contains no elements.
- Conclusion: It is a null set.
- (ii) Set of even prime numbers:
- Solution: The number 2 is an even prime number, so the set contains one element.
- Conclusion: It is not a null set.
- (iii) $\{x : x \text{ is a natural number, } x < 5 \text{ and } x > 7\}$:
- Solution: No natural number can be less than 5 and greater than 7 at the same time.
- Conclusion: It is a null set.
- (iv) $\{y : y \text{ is a point common to any two parallel lines}\}$:
- Solution: Parallel lines never intersect; therefore, they have no common point.
- Conclusion: It is a null set.
Q2. Which of the following sets are finite or infinite?
- Definition: A set which is empty or consists of a definite number of elements is called finite; otherwise, it is infinite.
- (i) The set of months of a year: Finite (There are exactly 12 months).
- (ii) $\{1, 2, 3, \dots\}$: Infinite (It contains an infinite number of natural numbers).
- (iii) $\{1, 2, 3, \dots, 99, 100\}$: Finite (It contains elements from 1 to 100).
- (iv) The set of positive integers greater than 100: Infinite (Numbers greater than 100 are endless).
- (v) The set of prime numbers less than 99: Finite (There is a countable number of prime numbers below 99).
Q3. State whether each of the following set is finite or infinite:
- (i) The set of lines which are parallel to the x-axis: Infinite (Infinite lines can be drawn parallel to the x-axis).
- (ii) The set of letters in the English alphabet: Finite (There are exactly 26 letters).
- (iii) The set of numbers which are multiple of 5: Infinite (Multiples of 5 are endless: 5, 10, 15, ...).
- (iv) The set of animals living on the earth: Finite (The number is large, but it is a fixed/countable number).
- (v) The set of circles passing through the origin (0,0): Infinite (Infinite circles of different radii can pass through the origin).
Q4. In the following, state whether $A = B$ or not:
- (i) $A = \{a, b, c, d\}; B = \{d, c, b, a\}$:
- Solution: The order of elements does not matter. Both have the same elements.
- Conclusion: $A = B$.
- (ii) $A = \{4, 8, 12, 16\}; B = \{8, 4, 16, 18\}$:
- Solution: $12 \in A$ but $12 \notin B$; also $18 \in B$ but $18 \notin A$.
- Conclusion: $A \neq B$.
- (iii) $A = \{2, 4, 6, 8, 10\}; B = \{x : x \text{ is a positive even integer and } x \le 10\}$:
- Solution: $B$ also contains $\{2, 4, 6, 8, 10\}$.
- Conclusion: $A = B$.
- (iv) $A = \{x : x \text{ is a multiple of 10}\}; B = \{10, 15, 20, 25, 30, \dots\}$:
- Solution: $A = \{10, 20, 30, \dots\}$. $B$ contains 15, 25, etc., which are not in $A$.
- Conclusion: $A \neq B$.
Q5. Are the following pair of sets equal? Give reasons.
- (i) $A = \{2, 3\}; B = \{x : x \text{ is a solution of } x^2 + 5x + 6 = 0\}$:
- Solution: Solving $x^2 + 5x + 6 = 0$ gives $x = -2, -3$. So $B = \{-2, -3\}$.
- Conclusion: $A \neq B$.
- (ii) $A = \{x : x \text{ is a letter in the word FOLLOW}\}; B = \{y : y \text{ is a letter in the word WOLF}\}$:
- Solution: $A = \{F, O, L, W\}$ and $B = \{W, O, L, F\}$.
- Conclusion: $A = B$.
Q6. From the sets given below, select 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: By comparing the elements of all sets :
- $B$ and $D$ have the same elements $\{1, 2, 3, 4\}$.
- $E$ and $G$ have the same elements $\{-1, 1\}$.
- Conclusion: The equal sets are $B = D$ and $E = G$.
Key Concepts Covered in Exercise 1.2
To solve this exercise effectively, students must understand these four pillars:
-
The Empty Set: Also known as the Null or Void set. It contains no elements and is denoted by the symbol $\phi$ or $\{ \}$.
-
Finite Sets: A set which is empty or consists of a definite (countable) number of elements.
-
Infinite Sets: A set where the process of counting elements does not come to an end (e.g., the set of all natural numbers).
-
Equal Sets: Two sets $A$ and $B$ are said to be equal if they have exactly the same elements, regardless of the order.
FAQs – Class 11 Maths Chapter 1 Exercise 1.2 Sets
Q1. What is the primary focus of Exercise 1.2?
The main focus is on identifying and classifying sets into categories like Empty, Finite, Infinite, and Equal sets based on their definitions.
Q2. How do we represent an Infinite Set in Roster Form?
Since we cannot list every element, we list a few elements to show the pattern followed by three dots (ellipses). For example: $\{1, 2, 3, \dots\}$.
Q3. Is the set $\{0\}$ an Empty Set?
No, $\{0\}$ is not an empty set. It is a singleton set because it contains one element: the number zero. An empty set is written as $\{ \}$ or $\phi$.
Q4. Why is Exercise 1.2 important for exam preparation?
Questions from this exercise often appear as Multiple Choice Questions (MCQs) or short-answer questions, testing a student's fundamental clarity on set theory.