Answer :

(i): A set of odd natural numbers divisible by $2$ is a null set because no odd number is divisible by $2$ .

(ii): A set of even prime numbers is not a null set because $2$ is an even prime number.

(iii): ${x$ : $x$ is a natural number, $x<5$ and $x>7}$ is a null set because a number cannot be simultaneously less than $5$ and greater than $7.$

(iv): ${y$ : $y$ is a point common to any two parallel lines $}$ is a null set because parallel lines do not intersect. Hence, they have no common point.

Answer :

(i): The set of months of a year is a finite set because it has $12$ elements.

(ii): ${1,2,3 \ldots}$ is an infinite set as it has infinite number of natural numbers.

(iii): ${1,2,3 \ldots 99,100}$ is a finite set because the numbers from $1$ to $100$ are finite in number.

(iv): The set of positive integers greater than $100$ is an infinite set because positive integers greater than $100$ are infinite in number.

(v): The set of prime numbers less than $99$ is a finite set because prime numbers less than $99$ are finite in number.

Answer :

(i): The set of lines which are parallel to the $x$-axis is an infinite set because lines parallel to the $x$-axis are infinite in number.

(ii): The set of letters in the English alphabet is a finite set because it has $26$ elements.

(iii): The set of numbers which are multiple of $5$ is an infinite set because multiples of $5$ are infinite in number.

(iv): The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (although it is quite a big number).

(v): The set of circles passing through the origin $(0,0)$ is an infinite set because infinite number of circles can pass through the origin.

Answer :

(i): $A={a, b, c, d} ; B={d, c, b, a}$

The order in which the elements of a set are listed is not significant.

$A=B$

(ii): $A={4,8,12,16} ; B={8,4,16,18}$

It can be seen that $12 \in A$ but $12 \notin B$.

$ \therefore \ \ A \neq B $

(iii): $A={2,4,6,8,10}$

$B={x: x$ is a positive even integer and $x \leq 10}$ $={2,4,6,8,10}$

$\therefore \ \ A=B$

(iv): $A={x: x$ is a multiple of $10 }$

$B={10,15,20,25,30 \ldots}$

It can be seen that $15 \in B$ but $15 \notin A$.

$\therefore \ \ A \neq B$

Answer :

(i): $A={2,3} ; B={x: x$ is a solution of $x^{2}+5 x+6=0}$

The equation $x^{2}+5 x+6=0$ can be solved as:

$x(x+3)+2(x+3)=0$

$(x+2)(x+3)=0$

$x=-2$ or $x=-3$

$\therefore \ \ A={2,3} ; B={-2,-3}$

$\therefore \ \ A \neq B$

(ii): $A={x: x$ is a letter in the word FOLLOW $}={F, O, L, W}$

$B={y: y$ is a letter in the word WOLF $}={W, O, L, F}$

The order in which the elements of a set are listed is not significant.

$\therefore \ \ A=B$

Answer :

$A={2,4,8,12} ; \ \ B={1,2,3,4} ; \ \ C={4,8,12,14}$

$D={3,1,4,2} ; \ \ E={-1,1} ; \ \ F={0, a}$

$G={1,-1} ; \ \ A={0,1}$

It can be seen that

$8 \in A, 8 \notin B, 8 \notin D, 8 \notin E, 8 \notin F, 8 \notin G, 8 \notin H$

$\Rightarrow A \neq B, A \neq D, A \neq E, A \neq F, A \neq G, A \neq H$

Also, $2 \in A, 2 \notin C$

$\therefore \ \ A \neq C$

$3 \in B, 3 \notin C, 3 \notin E, 3 \notin F, 3 \notin G, 3 \notin H$

$\therefore \ \ B \neq C, B \neq E, B \neq F, B \neq G, B \neq H$

$12 \in C, 12 \notin D, 12 \notin E, 12 \notin F, 12 \notin G, 12 \notin H$

$\therefore \ \ C \neq D, C \neq E, C \neq F, C \neq G, C \neq H 4$

$\in D, 4 \notin E, 4 \notin F, 4 \notin G, 4 \notin H$

$\therefore \ \ D \neq E, D \neq F, D \neq G, D \neq H$

Similarly, $E \neq F, E \neq G, E \neq H$

$F \neq G, F \neq H, G \neq H$

The order in which the elements of a set are listed is not significant.

$\therefore \ \ B=D$ and $E=G$

Hence, among the given sets, $B=D$ and $E=G$.