NCERT Solutions Class 11 Maths Chapter 1 Exercise 1.3

Q1: Fill in the symbols $\subset$ or $\not\subset$.

(i) $\{ 2, 3, 4 \} \dots \{ 1, 2, 3, 4, 5 \}$

All elements of the first set (2, 3, 4) are present in the second set.
Answer: $\subset$

(ii) $\{ a, b, c \} \dots \{ b, c, d \}$

Element ‘a’ is in the first set but not in the second.
Answer: $\not\subset$

(iii) $\{x : x \text{ is a student of Class XI}\} \dots \{x : x \text{ is a student of your school}\}$

All Class XI students are students of the school.
Answer: $\subset$

(iv) $\{x : x \text{ is a circle in plane}\} \dots \{x : x \text{ is a circle with radius 1}\}$

The first set includes all circles (radius 2, 5, etc.). The second only includes unit circles.
Answer: $\not\subset$

(v) $\{x : x \text{ is a triangle}\} \dots \{x : x \text{ is a rectangle}\}$

A triangle is never a rectangle.
Answer: $\not\subset$

(vi) $\{x : x \text{ is an equilateral triangle}\} \dots \{x : x \text{ is a triangle}\}$

All equilateral triangles are triangles.
Answer: $\subset$

(vii) $\{x : x \text{ is an even natural number}\} \dots \{x : x \text{ is an integer}\}$

Even natural numbers (2, 4, 6…) are integers.
Answer: $\subset$

Q2: Examine validity of statements.

(i) $\{ a, b \} \not\subset \{ b, c, a \}$

Every element of {a, b} is in {b, c, a}. So, it IS a subset.
Statement says “not a subset”.
Result: False

(ii) $\{ a, e \} \subset \{ x : x \text{ is a vowel}\}$

Vowels = {a, e, i, o, u}. {a, e} is contained in it.
Result: True

(iii) $\{ 1, 2, 3 \} \subset \{ 1, 3, 5 \}$

Element 2 is in the first set but not the second.
Result: False

(iv) $\{ a \} \subset \{ a, b, c \}$

{a} is a set containing element ‘a’. It is a subset.
Result: True

(v) $\{ a \} \in \{ a, b, c \}$

The symbol $\in$ relates an element to a set. Here, ‘a’ is the element, not ‘{a}’.
Result: False

(vi) $\{ x : x \text{ is even natural } < 6 \} \subset \{ x : x \text{ is natural number dividing 36} \}$

Set 1 = {2, 4}
Set 2 (Divisors of 36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}
{2, 4} is contained in Set 2.
Result: True

Q3: Which statements are incorrect?

Note: In set A, the elements are: $1$, $2$, $\{3, 4\}$, and $5$.
$\{3, 4\}$ is treated as a single element.

(i) $\{3, 4\} \subset A$ Incorrect
Reason: $\{3, 4\}$ is an element, so we use $\in$. If we want a subset, we need $\{\{3, 4\}\}$.

(ii) $\{3, 4\} \in A$ Correct

(iii) $\{\{3, 4\}\} \subset A$ Correct

(iv) $1 \in A$ Correct

(v) $1 \subset A$ Incorrect
Reason: 1 is an element, not a set.

(vi) $\{1, 2, 5\} \subset A$ Correct

(vii) $\{1, 2, 5\} \in A$ Incorrect
Reason: $\{1, 2, 5\}$ is a subset, not an element listed inside A.

(viii) $\{1, 2, 3\} \subset A$ Incorrect
Reason: 3 is not an element of A (it is inside the element {3,4}).

(ix) $\phi \in A$ Incorrect
Reason: The null set is a subset of every set, but not an element unless explicitly listed.

(x) $\phi \subset A$ Correct

(xi) $\{\phi\} \subset A$ Incorrect
Reason: This would require $\phi \in A$.

Q4: Write down all subsets.

(i) $\{a\}$: $\phi, \{a\}$
(ii) $\{a, b\}$: $\phi, \{a\}, \{b\}, \{a, b\}$
(iii) $\{1, 2, 3\}$: $\phi, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}$
(iv) $\phi$: $\phi$

Q5: Write as intervals.

(i) $\{x : x \in R, -4 < x \le 6\}$: $(-4, 6]$
(ii) $\{x : x \in R, -12 < x < -10\}$: $(-12, -10)$
(iii) $\{x : x \in R, 0 \le x < 7\}$: $[0, 7)$
(iv) $\{x : x \in R, 3 \le x \le 4\}$: $[3, 4]$

Q6: Write in set-builder form.

(i) $(-3, 0)$: $\{x : x \in R, -3 < x < 0\}$
(ii) $[6, 12]$: $\{x : x \in R, 6 \le x \le 12\}$
(iii) $(6, 12]$: $\{x : x \in R, 6 < x \le 12\}$
(iv) $[-23, 5)$: $\{x : x \in R, -23 \le x < 5\}$

Q7 & Q8: Universal Set Proposals.

Q7: Universal sets for Triangles

(i) Right triangles, (ii) Isosceles triangles.
Proposal: The set of all triangles (or the set of all 2D polygons).

Q8: Universal set for A={1,3,5}, B={2,4,6}, C={0,2,4,6,8}

The universal set must contain all elements from A, B, and C.
Combined elements: $\{0, 1, 2, 3, 4, 5, 6, 8\}$.

(i) $\{0…6\}$ (Missing 8) $\to$ No
(ii) $\phi$ $\to$ No
(iii) $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ $\to$ Yes (Contains all)
(iv) $\{1…8\}$ (Missing 0) $\to$ No

Answer: (iii)

Sets

1. Insert Subset Symbol ($\subset$ or $\not\subset$)

2. True or False Statements

3. Analysis of Set $A = \{ 1, 2, \{ 3, 4 }, 5 \}$

4, 5, 6. Subsets & Intervals

7, 8. Universal Sets