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) \( \{x : x \text{ is an even natural number}\} \ldots \{x : x \text{ is an integer}\} \)
(i) \( \subset \), (ii) \( \not\subset \), (iii) \( \subset \), (iv) \( \not\subset \), (v) \( \not\subset \), (vi) \( \subset \), (vii) \( \subset \)
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) \( \{x : x \text{ is an even natural number less than }6\} \subset \{x : x \text{ is a natural number which divides }36\} \)
(i) False, (ii) True, (iii) False, (iv) True, (v) False, (vi) True
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) \( \{\varphi\} \subset A \)
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 \(A\)).
Write down all the subsets of the following sets:
(i) \( \{a\} \)
(ii) \( \{a, b\} \)
(iii) \( \{1, 2, 3\} \)
(iv) \( \varphi \)
(i) \( \varphi, \{a\} \)
(ii) \( \varphi, \{a\}, \{b\}, \{a, b\} \)
(iii) \( \varphi, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\} \)
(iv) \( \varphi \)
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) \( \{x : x \in \mathbb{R}, 3 \le x \le 4\} \)
(i) \( (-4, 6] \)
(ii) \( (-12, -10) \)
(iii) \( [0, 7) \)
(iv) \( [3, 4] \)
Write the following intervals in set-builder form:
(i) \( (-3, 0) \)
(ii) \( [6, 12] \)
(iii) \( (6, 12] \)
(iv) \( [-23, 5) \)
(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) \( \{x : x \in \mathbb{R}, -23 \le x < 5\} \)
What universal set(s) would you propose for each of the following:
(i) The set of right triangles.
(ii) The set of isosceles triangles.
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) \( \{1, 2, 3, 4, 5, 6, 7, 8\} \)
(iii)