SETS-Exercise 1.3
Maths - Exercise
Q1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces :
(i) { 2, 3, 4 } . . . { 1, 2, 3, 4,5 }
(ii) { a, b, c } . . . { b, c, d }
(iii) {x : x is a student of Class XI of your school}. . .{x : x student of your school}
(iv) {x : x is a circle in the plane} . . .{x : x is a circle in the same plane with radius 1 unit}
(v) {x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}
(vi) {x : x is an equilateral triangle in a plane} . . . {x : x is a triangle in the same plane}
(vii) {x : x is an even natural number} . . . {x : x is an integer}
Solution
Solution: A set \(A\) is said to be a subset of a set \(B\) if every element of \(A\) is also an element of \(B\). Using this idea, we fill in the appropriate symbols in each case.
\( \begin{aligned} \text{(i)}\;& \{2, 3, 4\} \subset \{1, 2, 3, 4, 5\} \\ &\text{since every element of the first set belongs to the second set.} \end{aligned} \)
\( \begin{aligned} \text{(ii)}\;& \{a, b, c\} \not\subset \{b, c, d\} \\ &\text{because the element } a \text{ does not belong to the second set.} \end{aligned} \)
\( \begin{aligned} \text{(iii)}\;& \{x : x \text{ is a student of Class XI of your school}\} \subset \{x : x \text{ is a student of your school}\} \\ &\text{as every Class XI student of the school is also a student of the same school.} \end{aligned} \)
\( \begin{aligned} \text{(iv)}\;& \{x : x \text{ is a circle in the plane}\} \not\subset \{x : x \text{ is a circle in the same plane with radius } 1 \text{ unit}\} \\ &\text{because circles of any radius exist in the plane, not only those of radius } 1. \end{aligned} \)
\( \begin{aligned} \text{(v)}\;& \{x : x \text{ is a triangle in a plane}\} \not\subset \{x : x \text{ is a rectangle in the plane}\} \\ &\text{since no triangle can be a rectangle.} \end{aligned} \)
\( \begin{aligned} \text{(vi)}\;& \{x : x \text{ is an equilateral triangle in a plane}\} \subset \{x : x \text{ is a triangle in the same plane}\} \\ &\text{because every equilateral triangle is a triangle.} \end{aligned} \)
\( \begin{aligned} \text{(vii)}\;& \{x : x \text{ is an even natural number}\} \subset \{x : x \text{ is an integer}\} \\ &\text{as every even natural number is also an integer.} \end{aligned} \)
Q2. 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}
Solution
Solution: Each statement is examined using the definition of a subset and the meaning of the symbols \( \subset \), \( \not\subset \), and \( \in \).
\( \begin{aligned} \text{(i)}\;& \{a, b\} \not\subset \{b, c, a\}. \\ &\text{Both } a \text{ and } b \text{ belong to the set } \{b, c, a\}. \\ &\text{Hence, } \{a, b\} \subset \{b, c, a\}. \\ &\Rightarrow \text{The given statement is false.} \end{aligned} \)
\( \begin{aligned} \text{(ii)}\;& \{a, e\} \subset \{x : x \text{ is a vowel in the English alphabet}\}. \\ &\text{Both } a \text{ and } e \text{ are vowels.} \\ &\Rightarrow \text{The statement is true.} \end{aligned} \)
\( \begin{aligned} \text{(iii)}\;& \{1, 2, 3\} \subset \{1, 3, 5\}. \\ &\text{The element } 2 \text{ does not belong to } \{1, 3, 5\}. \\ &\Rightarrow \text{The statement is false.} \end{aligned} \)
\( \begin{aligned} \text{(iv)}\;& \{a\} \subset \{a, b, c\}. \\ &\text{The element } a \text{ belongs to the second set.} \\ &\Rightarrow \text{The statement is true.} \end{aligned} \)
\( \begin{aligned} \text{(v)}\;& \{a\} \in \{a, b, c\}. \\ &\text{The elements of the set } \{a, b, c\} \text{ are } a, b, \text{ and } c, \text{ not } \{a\}. \\ &\Rightarrow \text{The statement is false.} \end{aligned} \)
\( \begin{aligned} \text{(vi)}\;& \{x : x \text{ is an even natural number less than } 6\} \subset \{x : x \text{ is a natural number which divides } 36\}. \\ &\text{The first set is } \{2, 4\}. \\ &\text{Both } 2 \text{ and } 4 \text{ divide } 36. \\ &\Rightarrow \text{The statement is true.} \end{aligned} \)
Q3. 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) \(\phi\) ∈ A (x) \(\phi\) ⊂ A (xi) \(\{\phi\}\) ⊂ A
Solution
Solution: The given set is \( A = \{1, 2, \{3, 4\}, 5\} \). Here, the elements of \(A\) are \(1\), \(2\), the set \(\{3,4\}\), and \(5\). Care must be taken to distinguish between an element of a set and a subset of a set.
\( \begin{aligned} \text{(i)}\;& \{3, 4\} \subset A. \\ &\text{The elements of } \{3,4\} \text{ are } 3 \text{ and } 4, \text{ but neither } 3 \text{ nor } 4 \text{ is an element of } A. \\ &\Rightarrow \text{This statement is incorrect.} \end{aligned} \)
\( \begin{aligned} \text{(ii)}\;& \{3, 4\} \in A. \\ &\text{The set } \{3,4\} \text{ itself is an element of } A. \\ &\Rightarrow \text{This statement is correct.} \end{aligned} \)
\( \begin{aligned} \text{(iii)}\;& \{\{3, 4\}\} \subset A. \\ &\text{The only element of } \{\{3,4\}\} \text{ is } \{3,4\}, \text{ which belongs to } A. \\ &\Rightarrow \text{This statement is correct.} \end{aligned} \)
\( \begin{aligned} \text{(iv)}\;& 1 \in A. \\ &\text{The number } 1 \text{ is an element of } A. \\ &\Rightarrow \text{This statement is correct.} \end{aligned} \)
\( \begin{aligned} \text{(v)}\;& 1 \subset A. \\ &\text{The symbol } \subset \text{ is defined only for sets, and } 1 \text{ is not a set.} \\ &\Rightarrow \text{This statement is incorrect.} \end{aligned} \)
\( \begin{aligned} \text{(vi)}\;& \{1, 2, 5\} \subset A. \\ &\text{All the elements } 1, 2, \text{ and } 5 \text{ belong to } A. \\ &\Rightarrow \text{This statement is correct.} \end{aligned} \)
\( \begin{aligned} \text{(vii)}\;& \{1, 2, 5\} \in A. \\ &\text{The set } \{1,2,5\} \text{ is not listed as an element of } A. \\ &\Rightarrow \text{This statement is incorrect.} \end{aligned} \)
\( \begin{aligned} \text{(viii)}\;& \{1, 2, 3\} \subset A. \\ &\text{Although } 1 \text{ and } 2 \text{ belong to } A, \text{ the element } 3 \text{ does not belong to } A. \\ &\Rightarrow \text{This statement is incorrect.} \end{aligned} \)
\( \begin{aligned} \text{(ix)}\;& \varnothing \in A. \\ &\text{The empty set is not listed as an element of } A. \\ &\Rightarrow \text{This statement is incorrect.} \end{aligned} \)
\( \begin{aligned} \text{(x)}\;& \varnothing \subset A. \\ &\text{The empty set is a subset of every set.} \\ &\Rightarrow \text{This statement is correct.} \end{aligned} \)
\( \begin{aligned} \text{(xi)}\;& \{\varnothing\} \subset A. \\ &\text{This would require } \varnothing \in A, \text{ which is not true.} \\ &\Rightarrow \text{This statement is incorrect.} \end{aligned} \)
Thus, the incorrect statements are: (i), (v), (vii), (viii), (ix) and (xi).
Q4. Write down all the subsets of the following sets
(i) {a}
(ii){a, b}
(iii){1, 2, 3}
(iv) \(\varnothing\)
Solution
Solution: The subsets of a set include the empty set, the set itself, and all possible collections formed by choosing elements from the given set without repetition.
\( \begin{aligned} \text{(i)}\;& \{a\}. \\ &\text{The possible subsets are } \varnothing \text{ and } \{a\}. \\ &\Rightarrow \text{Subsets of } \{a\} \text{ are } \{\varnothing, \{a\}\}. \end{aligned} \)
\( \begin{aligned} \text{(ii)}\;& \{a, b\}. \\ &\text{The possible subsets are } \varnothing, \{a\}, \{b\}, \text{ and } \{a, b\}. \\ &\Rightarrow \text{Subsets of } \{a, b\} \text{ are } \{\varnothing, \{a\}, \{b\}, \{a, b\}\}. \end{aligned} \)
\( \begin{aligned} \text{(iii)}\;& \{1, 2, 3\}. \\ &\text{The subsets are } \varnothing, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}. \\ &\Rightarrow \text{All subsets of } \{1, 2, 3\} \text{ are } \{\varnothing, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}. \end{aligned} \)
\( \begin{aligned} \text{(iv)}\;& \varnothing. \\ &\text{The empty set has only one subset, which is the empty set itself.} \\ &\Rightarrow \text{Subsets of } \varnothing \text{ are } \{\varnothing\}. \end{aligned} \)
Q5. 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}
Solution
Solution: Each given set is described in set-builder form. We convert it into interval notation by observing whether the end points are included or excluded.
\( \begin{aligned} \text{(i)}\;& \{x : x \in \mathbb{R}, -4 < x \le 6\}. \\ &\text{The value } -4 \text{ is not included, while } 6 \text{ is included.} \\ &\Rightarrow \text{The interval is } (-4,\,6]. \end{aligned} \)
\( \begin{aligned} \text{(ii)}\;& \{x : x \in \mathbb{R}, -12 < x < -10\}. \\ &\text{Neither } -12 \text{ nor } -10 \text{ is included.} \\ &\Rightarrow \text{The interval is } (-12,\,-10). \end{aligned} \)
\( \begin{aligned} \text{(iii)}\;& \{x : x \in \mathbb{R}, 0 \le x < 7\}. \\ &\text{The value } 0 \text{ is included, while } 7 \text{ is not included.} \\ &\Rightarrow \text{The interval is } [0,\,7). \end{aligned} \)
\( \begin{aligned} \text{(iv)}\;& \{x : x \in \mathbb{R}, 3 \le x \le 4\}. \\ &\text{Both } 3 \text{ and } 4 \text{ are included.} \\ &\Rightarrow \text{The interval is } [3,\,4]. \end{aligned} \)
Q6. Write the following intervals in set-builder form :
(i) (– 3, 0)
(ii) [6 , 12]
(iii) (6, 12]
(iv) [–23, 5)
Solution
Solution: Each given interval is expressed in set-builder form by translating the interval notation into inequalities involving real numbers.
\( \begin{aligned} \text{(i)}\;& (-3,\,0). \\ &\text{The end points } -3 \text{ and } 0 \text{ are not included.} \\ &\Rightarrow \{x : x \in \mathbb{R},\; -3 < x < 0\}. \end{aligned} \)
\( \begin{aligned} \text{(ii)}\;& [6,\,12]. \\ &\text{Both end points } 6 \text{ and } 12 \text{ are included.} \\ &\Rightarrow \{x : x \in \mathbb{R},\; 6 \le x \le 12\}. \end{aligned} \)
\( \begin{aligned} \text{(iii)}\;& (6,\,12]. \\ &\text{The point } 6 \text{ is excluded, while } 12 \text{ is included.} \\ &\Rightarrow \{x : x \in \mathbb{R},\; 6 < x \le 12\}. \end{aligned} \)
\( \begin{aligned} \text{(iv)}\;& [-23,\,5). \\ &\text{The point } -23 \text{ is included, while } 5 \text{ is excluded.} \\ &\Rightarrow \{x : x \in \mathbb{R},\; -23 \le x < 5\}. \end{aligned} \)
Q7. What universal set(s) would you propose for each of the following :
(i) The set of right triangles.
(ii) The set of isosceles triangles.
Solution
Solution: A universal set must contain all the elements under discussion along with every set related to the given set. Keeping this in mind, we propose suitable universal sets for each case.
\( \begin{aligned} \text{(i)}\;& \text{The set of right triangles.} \\ &\text{Every right triangle is a triangle.} \\ &\Rightarrow \text{A suitable universal set is the set of all triangles in a plane.} \end{aligned} \)
\( \begin{aligned} \text{(ii)}\;& \text{The set of isosceles triangles.} \\ &\text{Every isosceles triangle is also a triangle.} \\ &\Rightarrow \text{A suitable universal set is the set of all triangles in a plane.} \end{aligned} \)
Q8. 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) \(\phi\)
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1,2,3,4,5,6,7,8}
Solution
Solution: A universal set must contain every element of the given sets \(A\), \(B\), and \(C\). Hence, each proposed set is examined to check whether it includes all elements of \(A = \{1,3,5\}\), \(B = \{2,4,6\}\), and \(C = \{0,2,4,6,8\}\).
\( \begin{aligned} \text{(i)}\;& \{0,1,2,3,4,5,6\}. \\ &\text{This set contains all elements of } A \text{ and } B, \text{ but } 8 \in C \text{ is missing.} \\ &\Rightarrow \text{It cannot be a universal set.} \end{aligned} \)
\( \begin{aligned} \text{(ii)}\;& \varnothing. \\ &\text{The empty set contains no elements.} \\ &\Rightarrow \text{It cannot contain the elements of } A, B, \text{ and } C. \\ &\Rightarrow \text{It cannot be a universal set.} \end{aligned} \)
\( \begin{aligned} \text{(iii)}\;& \{0,1,2,3,4,5,6,7,8,9,10\}. \\ &\text{This set contains every element of } A, B, \text{ and } C. \\ &\Rightarrow \text{It may be considered as a universal set.} \end{aligned} \)
\( \begin{aligned} \text{(iv)}\;& \{1,2,3,4,5,6,7,8\}. \\ &\text{The element } 0 \in C \text{ is not present in this set.} \\ &\Rightarrow \text{It cannot be a universal set.} \end{aligned} \)
Hence, option (iii) alone can be considered as a universal set for the sets \(A\), \(B\), and \(C\).
