NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.4 Sets introduces students to the practical arithmetic of set theory. While previous exercises focused on defining and categorizing sets, this exercise teaches students how to combine or compare sets using specific logical operations.
NCERT Solutions For Class 11 Maths Chapter 1 Sets Exercise 1.4
Q.
Which of the following are sets ? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-cricket batsmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.
Q.
In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?
Q.
Show that the following four conditions are equivalent:
(i) A ⊂ B
(ii) A – B = Φ
(iii) A ∪ B = B
(iv) A ∩ B = A
Q.
Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C
Q.
State whether each of the following statement is true or false. Justify your answer.
(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.
(ii) {a, e, i, o, u } and { a, b, c, d }are disjoint sets.
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.
Q.
Find the intersection of each pair of sets of
(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = [ a, e, i, o, u} B = {a, b, c}
(iii) A = {x: x is a natural number and multiple of 3}
B = {x: x is a natural number less than 6}
(iv) A = {x: x is a natural number and 1 < x ≤ 6}
B = {x: x is a natural number and 6 < x < 10}
(v) A = {1, 2, 3}, B = Φ
Q.
Find the union of each of the following pairs of sets:
(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = [ a, e, i, o, u} B = {a, b, c}
(iii) A = {x: x is a natural number and multiple of 3}
B = {x: x is a natural number less than 6}
(iv) A = {x: x is a natural number and 1 < x ≤ 6}
B = {x: x is a natural number and 6 < x < 10}
(v) A = {1, 2, 3}, B = Φ
Q.
In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers.
Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.
Q.
Assume that P (A) = P (B). Show that A = B
Q.
Are the following pair of sets equal? Give reasons.
(i) A = {2, 3},
B = {x: x is solution of x2 + 5x + 6 = 0}
(ii) A = {x: x is a letter in the word FOLLOW}
B = {y: y is a letter in the word WOLF}
Q.
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
Q.
Decide, among the following sets, which sets are subsets of one and another:
A = {x: x ∈ R and x satisfy x2 – 8x + 12 = 0},
B = {2, 4, 6}, C = {2, 4, 6, 8, . . .}, D = { 6 }.
Q.
In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
Q.
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
(i) (A ∪ B)′ = A′ ∩ B′
(ii) (A ∩ B)′ = A′ ∪ B′
Q.
Write the following as intervals:
(i) {x: x ∈ R, – 4 < x ≤ 6}
(ii) {x: x ∈ R, – 12 < x < –10}
(iii) {x: x ∈ R, 0 ≤ x < 7}
(iv) {x: x ∈ R, 3 ≤ x ≤ 4}
Q.
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}
Q.
Is it true that for any sets A and B, P (A )∪ P (B) = P (A ∪ B)? Justify your answer.
Prepared according to the latest CBSE Class 11 Maths syllabus, Exercise 1.4 focuses on Union, Intersection, and Difference of Sets. These operations are the mathematical equivalent of addition and subtraction and are essential for solving word problems and understanding probability.
Q1. Which of the following are sets? Justify your answer.
- The collection of all months of a year beginning with the letter J. * Ans: This is a set because it is a well-defined collection of items; a month belonging to this collection can be easily identified.
- The collection of ten most talented writers of India. * Ans: This is not a well-defined collection because the criteria for judging a writer's talent differ from one person to the next; therefore, it is not a set.
- A team of eleven best-cricket batsmen of the world. * Ans: This is not a well-defined collection because the criteria for judging a batsman's talent vary; thus, it does not constitute a set.
- The collection of all natural numbers less than 100. * Ans: This is a set because the collection is well-defined and numbers belonging to it are easily identified.
Exercise
Q1. Which of the following are examples of the null set?
- Set of odd natural numbers divisible by 2. * Ans: This is a null set because no odd number is divisible by 2.
- Set of even prime numbers. * Ans: This is not a null set because 2 is an even prime number.
- $\{x : x \text{ is a natural number, } x < 5 \text{ and } x > 7\}$. * Ans: This is a null set because a number cannot be simultaneously less than 5 and greater than 7.
Q2. State whether the following sets are finite or infinite.
- The set of months of a year. * Ans: Finite set because it has exactly 12 elements.
- $\{1, 2, 3, ...\}$. * Ans: Infinite set as it has an infinite number of natural numbers.
- The set of prime numbers less than 99. * Ans: Finite set because prime numbers less than 99 are finite in number.
Exercise
Q1. Fill in the blanks with the symbols $\subset$ (subset) or $\not\subset$ (not a subset):
- $\{2, 3, 4\} \dots \{1, 2, 3, 4, 5\}$ * Ans: $\{2, 3, 4\} \subset \{1, 2, 3, 4, 5\}$.
- $\{a, b, c\} \dots \{b, c, d\}$ * Ans: $\{a, b, c\} \not\subset \{b, c, d\}$.
Q2. Write the following as intervals:
- $\{x : x \in R, -4 < x \le 6\}$ * Ans: $(-4, 6]$.
- $\{x : x \in R, 3 \le x \le 4\}$ * Ans: $[3, 4]$.
Exercise
Q1. Find the union ($A \cup B$) of each of the following pairs of sets:
- $X = \{1, 3, 5\}, Y = \{1, 2, 3\}$ * Ans: $X \cup Y = \{1, 2, 3, 5\}$.
- $A = \{1, 2, 3\}, B = \emptyset$ * Ans: $A \cup B = \{1, 2, 3\}$.
Q2. Find the intersection ($A \cap B$) of each pair of sets:
- $X = \{1, 3, 5\}, Y = \{1, 2, 3\}$ * Ans: $X \cap Y = \{1, 3\}$.
- $A = \{a, e, i, o, u\}, B = \{a, b, c\}$ * Ans: $A \cap B = \{a\}$.
Exercise Highlights
Q1. Show that the following conditions are equivalent: (i) $A \subset B$, (ii) $A - B = \emptyset$, (iii) $A \cup B = B$, (iv) $A \cap B = A$.
- Ans: The document provides step-by-step logical proofs showing that each condition implies the others, confirming they are equivalent.
Q2. Show that $A \cap B = A \cap C$ need not imply $B = C$.
- Ans: Let $A = \{0, 1\}$, $B = \{0, 2, 3\}$, and $C = \{0, 4, 5\}$. Here, $A \cap B = \{0\}$ and $A \cap C = \{0\}$, but $B \ne C$.
Key Concepts in Exercise 1.4
To master this exercise, you must become familiar with these three primary operations:
-
Union of Sets ($A \cup B$): The set containing all elements that are in $A$, or in $B$, or in both. It represents the "total" collection.
-
Intersection of Sets ($A \cap B$): The set containing only the elements that are common to both $A$ and $B$. If the intersection is empty ($\phi$), the sets are called Disjoint Sets.
-
Difference of Sets ($A - B$): The set of elements which belong to $A$ but not to $B$. Note that $A - B$ is usually not the same as $B - A$.
FAQs – Class 11 Maths Chapter 1 Exercise 1.4 Sets
Q1. What is the main goal of Exercise 1.4?
The goal is to perform operations on two or more sets and represent the results using both Roster form and Venn diagrams.
Q2. What are Disjoint Sets?
Two sets $A$ and $B$ are disjoint if they have no common elements, meaning $A \cap B = \phi$. In a Venn diagram, their circles would not overlap.
Q3. Does the order matter in Union and Intersection?
No. Both Union and Intersection are Commutative, meaning $A \cup B = B \cup A$ and $A \cap B = B \cap A$. However, this is not true for the Difference of sets.
Q4. How is the "Complement of a Set" related to this exercise?
While Exercise 1.4 focuses on Union and Intersection, the concept of Difference ($U - A$) leads directly into the "Complement of a Set" covered in the next section.