NCERT Solutions
Class 11 - Mathematics - Chapter 1: SETS
EXERCISE 1.3

Open NCERT PDFOpen NCERT Answers PDF

Question. 1

Make correct statements by filling in the symbols \( \subset \) or \( \not\subset \) in the blank spaces:

(i) \( \{2, 3, 4\} \ldots \{1, 2, 3, 4, 5\} \)
(ii) \( \{a, b, c\} \ldots \{b, c, d\} \)
(iii) \( \{x : x \text{ is a student of Class XI of your school}\} \ldots \{x : x \text{ is a student of your school}\} \)
(iv) \( \{x : x \text{ is a circle in the plane}\} \ldots \{x : x \text{ is a circle in the same plane with radius }1\text{ unit}\} \)
(v) \( \{x : x \text{ is a triangle in a plane}\} \ldots \{x : x \text{ is a rectangle in the plane}\} \)
(vi) \( \{x : x \text{ is an equilateral triangle in a plane}\} \ldots \{x : x \text{ is a triangle in the same plane}\} \)
(vii) \( \{x : x \text{ is an even natural number}\} \ldots \{x : x \text{ is an integer}\} \)

Answer:

(i) \( \subset \), (ii) \( \not\subset \), (iii) \( \subset \), (iv) \( \not\subset \), (v) \( \not\subset \), (vi) \( \subset \), (vii) \( \subset \)

Question. 2

Examine whether the following statements are true or false:

(i) \( \{a, b\} \subset \{b, c, a\} \)
(ii) \( \{a, e\} \subset \{x : x \text{ is a vowel in the English alphabet}\} \)
(iii) \( \{1, 2, 3\} \subset \{1, 3, 5\} \)
(iv) \( \{a\} \subset \{a, b, c\} \)
(v) \( \{a\} \in \{a, b, c\} \)
(vi) \( \{x : x \text{ is an even natural number less than }6\} \subset \{x : x \text{ is a natural number which divides }36\} \)

Answer:

(i) False, (ii) True, (iii) False, (iv) True, (v) False, (vi) True

Question. 3

Let \( A = \{1, 2, \{3, 4\}, 5\} \). Which of the following statements are incorrect and why?

(i) \( \{3, 4\} \subset A \)
(ii) \( \{3, 4\} \in A \)
(iii) \( \{\{3, 4\}\} \subset A \)
(iv) \( 1 \in A \)
(v) \( 1 \subset A \)
(vi) \( \{1, 2, 5\} \subset A \)
(vii) \( \{1, 2, 5\} \in A \)
(viii) \( 3 \in A \)
(ix) \( \varphi \in A \)
(x) \( \varphi \subset A \)
(xi) \( \{\varphi\} \subset A \)

Answer:

The incorrect statements are:

(i) \( \{3, 4\} \subset A \), because \( \{3, 4\} \in A \) (\(3\) and \(4\) are not elements of \(A\)).

(v) \( 1 \subset A \), because \( 1 \in A \).

(vii) \( \{1, 2, 5\} \in A \), because \( \{1, 2, 5\} \subset A \) but is not an element of \(A\).

(viii) \( 3 \in A \), because \( 3 \notin A \).

(ix) \( \varphi \in A \), because only \( \varphi \subset A \).

(xi) \( \{\varphi\} \subset A \), because \( \varphi \notin A \) (so \( \{\varphi\} \) is not a subset of \(A\)).

Question. 4

Write down all the subsets of the following sets:

(i) \( \{a\} \)
(ii) \( \{a, b\} \)
(iii) \( \{1, 2, 3\} \)
(iv) \( \varphi \)

Answer:

(i) \( \varphi, \{a\} \)

(ii) \( \varphi, \{a\}, \{b\}, \{a, b\} \)

(iii) \( \varphi, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\} \)

(iv) \( \varphi \)

Question. 5

Write the following as intervals:

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

Answer:

(i) \( (-4, 6] \)

(ii) \( (-12, -10) \)

(iii) \( [0, 7) \)

(iv) \( [3, 4] \)

Question. 6

Write the following intervals in set-builder form:

(i) \( (-3, 0) \)
(ii) \( [6, 12] \)
(iii) \( (6, 12] \)
(iv) \( [-23, 5) \)

Answer:

(i) \( \{x : x \in \mathbb{R}, -3 < x < 0\} \)

(ii) \( \{x : x \in \mathbb{R}, 6 \le x \le 12\} \)

(iii) \( \{x : x \in \mathbb{R}, 6 < x \le 12\} \)

(iv) \( \{x : x \in \mathbb{R}, -23 \le x < 5\} \)

Question. 7

What universal set(s) would you propose for each of the following:

(i) The set of right triangles.
(ii) The set of isosceles triangles.

Question. 8

Given the sets \( A = \{1, 3, 5\} \), \( B = \{2, 4, 6\} \) and \( C = \{0, 2, 4, 6, 8\} \), which of the following may be considered as universal set(s) for all the three sets \(A, B\) and \(C\)?

(i) \( \{0, 1, 2, 3, 4, 5, 6\} \)
(ii) \( \varphi \)
(iii) \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
(iv) \( \{1, 2, 3, 4, 5, 6, 7, 8\} \)

Answer:

(iii)