Exercise 1.2 Class 11 Maths Solutions

Given: Set of odd natural numbers divisible by 2.
As we know set is a collection of well - defined objects.
Let we represent the given set in roaster form:
\( \Rightarrow \) Set of odd natural numbers divisible by 2 is \( \{\phi\} \).
Because no odd natural number can be divided by 2. Hence, it is a null set.

Given: Set of even prime numbers.
As we know set is a collection of well-defined objects.
Let we represent the given set in roaster form:
\( \Rightarrow \) Set of even prime numbers is \( \{2\} \).
Because 2 is an even prime number. Hence, it is not a null set.

Given: \( \{x: x \) is a natural number, \( x < 5 \) and \( x > 7\} \)
As we know set is a collection of well-defined objects.
Let we represent the given set in roaster form
\( \Rightarrow\{x: x \) is a natural number, \( x < 5 \) and \( x > 7\} \) is \( \{\phi\} \).
Because no number can be simultaneously less than 5 and greater than 7. Hence, it is a null set.

Given: \( \{y: y \) is a point common to any two parallel lines}
As we know set is a collection of well-defined objects.
Let we represent the given set in roaster form:
\( \Rightarrow\{y: y \) is a point common to any two parallel lines \( \} \) is \( \{\phi\} \).
Because two parallel lines never meet at any of the point so they don't have any common point. Hence, it is a null set.

Given: The set of months of a year.
As we know set is a collection of well-defined objects.
Let we represent the given set in roaster form:
\( \Rightarrow \) Set of months of a year is {January, February, March, April, May, June, July, August, September, October, November, December}
Because the set contain 12 elements. Hence, it is a finite set.

Given: \( \{1,2,3, \ldots\} \).
As we know set is a collection of well-defined objects.
As it is already represented in roaster form:
\(\Rightarrow \text { Set }=\{1,2,3, \ldots\}\)
Because the set contain infinite number of natural numbers. Hence, it is an infinite set.

Given: \( \{1,2,3, \ldots 99,100\} \).
As we know set is a collection of well-defined objects.
As it is already represented in roaster form
\(\Rightarrow \text { Set }=\{1,2,3, \ldots 99,100\}\)
Because the set contain finite number from 1 to 100. Hence, it is a finite set.

Given: The set of positive integers greater than 100 .
As we know set is a collection of well-defined objects.
Let we represent the set in roaster form:
\( \Rightarrow \) Set of positive integers greater than \( 100=\{100,101,102 ,\}.\)
Because the set contain an infinite number from 100 to infinity. Hence, it is an infinite set.

Given: The set of prime numbers less than 99.
As we know set is a collection of well-defined objects.
Let we represent the set in roaster form:
\( \Rightarrow \) The set of prime numbers less than \( 99=\{2,3, \ldots 99 \} \).
Because the set contain finite prime number from 2 to 99 . Hence, it is a finite set.

Given: The set of lines which are parallel to the x-axis.
As we know set is a collection of well-defined objects.
Let we represent the set in set builder form:
\( \Rightarrow S=\{x: x \) is number of parallel lines to x-axis \( \} \).
Because the set of lines parallel to \( x \)-axis are infinite in number. Hence, it is an infinite set.

Given: The set of letters in the English alphabet.
As we know set is a collection of well-defined objects.
Let we represent the set in roaster form:
\( \Rightarrow \) \( \text{The set of letters in the English alphabet}=\{\mathrm{A}, \mathrm{B}, \mathrm{C}, \ldots, z\} \)
Because the set contain finite alphabet series and having 26 elements. Hence, it is a finite set.

Given: The set of numbers which are multiple of 5
As we know set is a collection of well-defined objects.
Let we represent the set in roaster form:
\( \Rightarrow \) \( \text{The set of numbers which are multiple of} \ 5=\{5,10,15, \ldots\} \).
Because the set contain infinite numbers which are multiple of 5. Hence, it is an infinite set.

Given: The set of animals living on the earth.
As we know set is a collection of well-defined objects
Let we represent the set in set builder form:
\( \Rightarrow \mathrm{S}=\{x: x \) is the set of animals living on the earth \( \} \).
Because the number of animal living on earth are though too large but they are finite in number. Hence, it is finite set.

Given: The set of circles passing through the origin \( (0,0) \).
As we know set is a collection of well-defined objects.
Let we represent the set in set builder form:
\( \Rightarrow \mathrm{S}=\{x: x \) is the set of circles passing through the origin \( (0,0)\} \).
Because the number of circles passing through the origin are infinite in number. Hence, it is an infinite set.

Given: \( \mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\} \ \mathrm{B}=\{\mathrm{d}, \mathrm{c}, \mathrm{b}, \mathrm{a}\} \).
Two set \( A \) and \( B \) are said to be equal if they have exactly same elements then we say \( A=B \).
Because elements of set A and B do not have significant order but A and \( B \) have same element.
\( \therefore \mathrm{A}=\mathrm{B} \).

Given: \( \mathrm{A}=\{4,8,12,16\} \mathrm{B}=\{8,4,16,18\} \).
Two set \( A \) and \( B \) are said to be equal if they have exactly same elements then we say \( A=B \).
As \( 12 \in \) A but 12 does not belongs to \( B \).
Because elements of set A and B do not have same element
\( \therefore \mathrm{A} \neq \mathrm{B} \).

Given: \( \mathrm{A}=\{2,4,6,8,10\} \mathrm{B}=\{x \) : x is positive even integer and \( x \leq \) \( 10\} \).
Let we represent the set B in roaster form:
\( \Rightarrow x \) is positive even integer and \( x \leq 10=\{2,4,6,8,10\} \).
Two set \( A \) and \( B \) are said to be equal if they have exactly same elements then we say \( A=B \).
Because elements of set \( A \) and \( B \) have same element.
\( \therefore \mathrm{A}=\mathrm{B} \).

Given: \( \mathrm{A}=\{x \) : x is a multiple of 10\( \}, \mathrm{B}=\{10,15,20,25,30, \ldots\} \)
Let we represent the set A in roaster form:
\( \Rightarrow \) Set \( \mathrm{A}=x \) is a multiple of \( 10=\{10,20,30, \ldots\} \).
And set \( \mathrm{B}=\{10,15,20,25,30, \ldots\} \)
Two set \( A \) and \( B \) are said to be equal if they have exactly same elements then we say \( A=B \).
As \( 15 \in \) B but 15 does not belongs to
Because elements of set A and B do not have same element.
\( \therefore \mathrm{A} \neq \mathrm{B} \).

\( \text{Given} : \mathrm{A}=\{2,3\}, \mathrm{B}=\left\{x: x\right. \) \( \text{is solution of} \left.x^{2}+5 x+6=0\right\} \)
Solution of equation \( x^{2}+5 x+6=0 \)
\(\Rightarrow x^{2}+3 x+2 x+6=0\)
\(\Rightarrow x(x+3)+2(x+3)=0\)
\(\Rightarrow(x+3)(x+2)=0\)
\(\Rightarrow x=\{-3,-2\}\)
Let we represent the set \( B \) in roaster form:
\( \Rightarrow \) Set \( \mathrm{B}=\left\{x\right. \) is solution of \( \left.x^{2}+5 x+6=0\right\}=\{-3,-2\} \).
And set \( \mathrm{A}=\{2,3\} \)
Two set A and B are said to be equal if they have exactly same elements then we say \( A=B \).
Because the elements of set A and B do not have same numbers.
\(\therefore \mathrm{A} \neq \mathrm{B} \text {. }\)

\( \text{Given} : \mathrm{A}=\{x: x \) is a letter in the word FOLLOW \( \}, \mathrm{B}=\{y: y \) is a letter in the word WOLF \( \} \).
Let we represent the set A in roaster form:
\( \Rightarrow \) Set \( A=\{x: x \) is a letter in the word FOLLOW \( \}=\{F, O, L, W\} \).
Let we represent the set \( B \) in roaster form:
\( \Rightarrow \) Set \( \mathrm{B}=\{y: y \) is a letter in the word WOLF \( \}=\{\mathrm{W}, \mathrm{O}, \mathrm{L}, \mathrm{F}\} \).
Two set A and B are said to be equal if they have exactly same elements then we say \( A=B \).
Because elements of set A and B do not have significant order, but A and \( B \) have same element.
\(\therefore \mathrm{A}=\mathrm{B} \text {. }\)

Given: \( \mathrm{A}=\{2,4,8,12\}, \mathrm{B}=\{1,2,3,4\}, \mathrm{C}=\{4,8,12,14\}, \mathrm{D}=\{3,1 \), \( 4,2\} \)
\(E=\{-1,1\}, F=\{0, a\}, G=\{1,-1\}, H=\{0,1\}\)
As we see,
\( 8 \in \mathrm{A} \), but 8 does not belong to \( \mathrm{B}, \mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G} \) and H
\(\Rightarrow A \neq B, A \neq D, A \neq E, A \neq F, A \neq G, A \neq H\)
And \( 2 \in \) A but 2 does not belong to \( C \)
\(\Rightarrow \mathrm{A} \neq \mathrm{C}\)
Now, \( 3 \in B \), but 3 does not belong to \( \mathrm{C}, \mathrm{E}, \mathrm{F}, \mathrm{G} \) and H
\(\Rightarrow B \neq C, B \neq E, B \neq F, B \neq G, B \neq H\)
Also, \( 12 \in \mathrm{C} \), but 12 does not belong to \( \mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G} \) and H
\(\Rightarrow C \neq D, C \neq E, C \neq F, C \neq G, C \neq H\)
Also, \( 4 \in \mathrm{D} \), but 4 does not belong to \( \mathrm{E}, \mathrm{F}, \mathrm{G} \) and H
\(\Rightarrow \mathrm{D} \neq \mathrm{E}, \mathrm{D} \neq \mathrm{F}, \mathrm{D} \neq \mathrm{G}, \mathrm{D} \neq \mathrm{H}\)
Also, \( -1 \in \) E, but -1 does not belong to \( F, G \) and \( H \)
\(\Rightarrow \mathrm{E} \neq \mathrm{F}, \mathrm{E} \neq \mathrm{G}, \mathrm{E} \neq \mathrm{H}\)
Also, a \( \in \) F, but a does not belong to G and H
\(\Rightarrow \mathrm{F} \neq \mathrm{G}, \mathrm{F} \neq \mathrm{H}\)
Also, \( -1 \in \mathrm{G} \), but -1 does not belong to H
\(\Rightarrow \mathrm{G} \neq \mathrm{H}\)
But \( \mathrm{B}=\mathrm{D} \) and \( \mathrm{E}=\mathrm{G} \).
As, two set A and B are said to be equal if they have exactly same elements then we say \( \mathrm{A}=\mathrm{B} \).
Because elements of set (B and D) and (E and G) do not have significant order but ( B and D ) and ( E and G ) have same element.
\( \therefore \mathrm{B}=\mathrm{D} \) and \( \mathrm{E}=\mathrm{G} \).