Chapter 1 Sets EXERCISE 1.3
EXERCISE 1.3
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): $\{x: x$ is a student of Class XI of your school $\} \ldots\{x: x$ student of your school $\}$
(iv): $\{x: x$ is a circle in the plane $\} \ldots\{x: x$ is a circle in the same plane with radius 1 unit $\}$
(v): $\{x: x$ is a triangle in a plane $\} \ldots\{x: x$ is a rectangle in the plane $\}$
(vi): $\{x: x$ is an equilateral triangle in a plane $\} \ldots\{x: x$ is a triangle in the same plane $\}$
(vii): $\{x: x$ is an even natural number $\} \ldots\{x: x$ is an integer $\}$
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Answer :
(i): $\{2,3,4\} \subset\{1,2,3,4,5\}$
(ii): $\{a, b, c\} \not \subset\{b, c, d\}$
(iii): $\{x$ : $x$ is a student of class XI of your school $\} \subset $ $\{x: x$ is student of your school $\}$
(iv): $\{x: x$ is a circle in the plane $ \} \not \subset \{x: x$ is a circle in the same plane with radius 1 unit $\}$
(v): $\{x: x$ is a triangle in a plane $ \} \not \subset \{x: x $ is a rectangle in the plane $\}$
(vi): $\{x: x$ is an equilateral triangle in a plane $\} \subset \{x: x$ in a triangle in the same plane $\}$
(vii): $\{x: x$ is an even natural number $\} \subset \{x: x$ is an integer $\}$
2. Examine whether the following statements are true or false:
(i): $\{a, b\} \not \subset\{b, c, a\}$
(ii): $\{a, e\} \subset\{x: x$ 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$ is an even natural number less than $6$ $\} \subset \{x: x$ is a natural number which divides 36 $ \}$
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Answer :
(i): False. Each element of $\{a, b\}$ is also an element of $\{b, c, a\}$.
(ii): True. $a, e$ are two vowels of the english alphabet.
(iii): False. $2 \in\{1,2,3\}$; however, $2 \notin\{1,3,5\}$
(iv): True. Each element of $\{a\}$ is also an element of $\{a, b, c\}$.
(v): False. The elements of $\{a, b, c\}$ are $a, b, c$. Therefore, $\{a\} \subset\{a, b, c\}$
(vi): True
$ \{x: x \text{ is an even natural number less than 6 }\} = \{2,4\}$
$ \{x:x \text{ is a natural number which divides 36 } \} = \{1,2,3,4,6,9,12,18,36\}$
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): $\{1,2,3\} \subset A$
(ix): $\phi \in A$
(x): $\phi \subset A$
(xi): $\{\phi\} \subset A$
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Answer :
$A=\{1,2,\{3,4\}, 5\}$
(i): The statement $\{3,4\} \subset A$ is incorrect because $3 \in\{3,4\}$; however, $3 \notin A$.
(ii): The statement $\{3,4\} \in A$ is correct because $\{3,4\}$ is an element of $A$.
(iii): The statement $\{\{3,4\}\} \subset A$ is correct because $\{3,4\} \in\{\{3,4\}\}$ and $\{3,4\} \in A$.
(iv): The statement $1 \in A$ is correct because $1$ is an element of $A$.
(v): The statement $1\subset A$ is incorrect because an element of a set can never be a subset of itself.
(vi): The statement $\{1,2,5\} \subset A$ is correct because each element of $\{1,2,5\}$ is also an element of $A.$
(vii): The statement $\{1,2,5\} \in A$ is incorrect because $\{1,2,5\}$ is not an element of $A$.
(viii): The statement $\{1,2,3\} \subset A$ is incorrect because $3 \in\{1,2,3\}$; however, $3 \notin A$.
(ix): The statement $\Phi \in A$ is incorrect because $\Phi$ is not an element of $A$.
(x): The statement $\Phi \subset A$ is correct because $\Phi$ is a subset of every set.
(xi): The statement $\{\Phi\} \subset A$ is incorrect because $\Phi \in\{\Phi\}$; however, $\Phi \in A$.
4. Write down all the subsets of the following sets
(i): $\{a\}$
(ii): $\{a, b\}$
(iii): $\{1,2,3\}$
(iv): $\phi$
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Answer :
(i): The subsets of $\{a\}$ are $\Phi$ and $\{a\}$.
(ii): The subsets of $\{a, b\}$ are $\Phi,\{a\},\{b\}$, and $\{a, b\}$.
(iii): The subsets of $\{1,2,3\}$ are $\Phi,\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\}$, and $\{1,2,3\}$
5. Write the following as intervals :
(i): $\{x: x \in R,-4<x \leq 6\}$
(ii): $\{x: x \in R,-12<x<-10\}$
(iii): $\{x: x \in R, 0 \leq x<7\}$
(iv): $\{x: x \in R, 3 \leq x \leq 4\}$
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Answer :
(i): $\{x: x \in R,-4<x \leq 6\}=(-4,6]$
(ii): $\{x: x \in R,-12<x<-10\}=(-12,-10)$
(iii): $\{x: x \in R, 0 \leq x<7\}=[0,7)$
(iv): $\{x : x \in$ $R, 3 \leq x \leq 4 \}=[3,4]$
6. Write the following intervals in set-builder form :
(i): $(-3,0)$
(ii): $[6,12]$
(iii): $(6,12]$
(iv): $[-23,5)$
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Answer :
(i): $(-3,0)=\{x: x \in R,-3<x<0\}$
(ii): $[6,12]=\{x: x \in R, 6 \leq x \leq 12\}$
(iii): $(6,12]=\{x: x \in R, 6<x \leq 12\}$
(iv): $[-23,5)=\{x: x \in R,-23 \leq x<5\}$
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.
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Answer :
(i): For the set of right triangles, the universal set can be the set of triangles or the set of polygons.
(ii): For the set of isosceles triangles, the universal set can be the set of triangles or the set of polygons or the set of two-dimensional figures.
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): $\phi$
(iii): $\{0,1,2,3,4,5,6,7,8,9,10\}$
(iv): $\{1,2,3,4,5,6,7,8\}$
Show Answer
Answer :
(i): It can be seen that $A \subset\{0,1,2,3,4,5,6\}$
$B \subset\{0,1,2,3,4,5,6\}$
However, C $\not \subset$ $\{0,1,2,3,4,5,6\}$
Therefore, the set $\{0,1,2,3,4,5,6\}$ cannot be the universal set for the sets $A, B,$ and $C.$
(ii): $ \ \ A \not \subset \phi, B$ $ \not \subset $ $\Phi, C \not \subset \Phi$
Therefore, $\Phi$ cannot be the universal set for the sets $A, B,$ and $C.$
(iii): $ \ \ \ A\quad \subset\{0,1,2,3,4,5,6,7,8,9,10\}$
$\qquad \ B$ $\quad \subset\{0,1,2,3,4,5,6,7,8,9,10\}$
$\qquad \ C$ $\quad\subset\{0,1,2,3,4,5,6,7,8,9,10\}$
Therefore, the set $\{0,1,2,3,4,5,6,7,8,9,10\}$ is the universal set for the sets $A, B,$ and $C.$
(iv): $ \ \ A \quad\subset\{1,2,3,4,5,6,7,8\}$
$ \qquad B \quad\subset\{1,2,3,4,5,6,7,8\}$
However, $ C$ $\quad\not \subset$ $\{1,2,3,4,5,6,7,8\}$
Therefore, the set $\{1,2,3,4,5,6,7,8\}$ cannot be the universal set for the sets $A, B,$ and $C.$
BETA
