Sets

(A set which does not contain any element is the null set $\phi$.)

(i) Set of odd natural numbers divisible by 2

Sol. Every number divisible by $2$ is even, so no odd natural number is divisible by $2$. The set has no element.

∴ It is a null set.

(ii) Set of even prime numbers

Sol. $2$ is an even prime number, so the set is $\{2\}$.

∴ It is not a null set.

(iii) $\{x:x$ is a natural numbers$,\ x<5$ $\text{and }x>7\}$

Sol. No natural number is less than $5$ and at the same time greater than $7$. The set has no element.

∴ It is a null set.

(iv) $\{y:y$ is a point common to any two parallel $\text{lines}\}$

Sol. Two parallel lines never meet, so they have no common point. The set has no element.

∴ It is a null set.

(A set which is empty or has a definite number of elements is finite; otherwise it is infinite.)

(i) The set of months of a year

Sol. A year has $12$ months, i.e., a definite number of months.

∴ The set is finite.

(ii) $\{1,2,3,\ldots\}$

Sol. $\{1,2,3,\ldots\}$ is the set of all natural numbers, which never ends.

∴ The set is infinite.

(iii) $\{1,2,3,\ldots,99,100\}$

Sol. The set has $100$ elements, i.e., a definite number of elements.

∴ The set is finite.

(iv) The set of positive integers greater than 100

Sol. The positive integers greater than $100$ are $101,102,103,\ldots$, which never end.

∴ The set is infinite.

(v) The set of prime numbers less than 99

Sol. There is only a definite number of prime numbers less than $99$ (they are $25$ in number).

∴ The set is finite.

(i) The set of lines which are parallel to the $x$-axis

Sol. There are infinitely many lines parallel to the x-axis.

∴ The set is infinite.

(ii) The set of letters in the English alphabet

Sol. The English alphabet has $26$ letters, i.e., a definite number of letters.

∴ The set is finite.

(iii) The set of numbers which are multiple of 5

Sol. The multiples of $5$ are $5,10,15,20,\ldots$, which never end.

∴ The set is infinite.

(iv) The set of animals living on the earth

Sol. The number of animals living on the earth is finite, although it is very large.

∴ The set is finite.

(v) The set of circles passing through the origin $(0,0)$

Sol. Infinitely many circles can pass through the origin (0, 0).

∴ The set is infinite.

(Two sets are equal if they have exactly the same elements; the order of the elements does not matter.)

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

Sol. $A=\{a,b,c,d\}$, $B=\{d,c,b,a\}$

A and B have exactly the same elements, though not in the same order.

∴ $A=B$

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

Sol. $A=\{4,8,12,16\}$, $B=\{8,4,16,18\}$

Since $12\in A$ but $12\notin B$,

∴ $A\ne B$

(iii) $A=\{2,4,6,8,10\}$, $B=\{x:x$ is positive even integer $\text{and }x\le10\}$

Sol. $A=\{2,4,6,8,10\}$

$B=\{x:x$ is a positive even integer $\text{and }x\le10\}$

The positive even integers less than or equal to 10 are $2,4,6,8,10$.

∴ $B=\{2,4,6,8,10\}=A$

Hence, $A=B$

(iv) $A=\{x:x\text{ is a multiple of }10\}$, $B=\{10,15,20,25,30,\ldots\}$

Sol.

Here, $15\in B$ but $15\notin A$, as 15 is not a multiple of 10.

∴ $A\ne B$

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

Sol. Solving $x^2+5x+6=0$,

So, $B=\{-2,-3\}$ while $A=\{2,3\}$. Here, $2\in A$ but $2\notin B$.

∴ $A$ and $B$ are not equal sets.

(ii) $A=\{x:x$ is a letter in the word $\text{FOLLOW}\}$, $B=\{y:y$ is a letter in the word $\text{WOLF}\}$

Sol. The distinct letters of FOLLOW are F, O, L, W, so $A=\{F,O,L,W\}$.

The letters of WOLF are W, O, L, F, so $B=\{W,O,L,F\}$.

$A$ and $B$ have exactly the same elements, though not in the same order.

∴ $A$ and $B$ are equal sets.

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

$E=\{-1,1\}$, $F=\{0,a\}$, $G=\{1,-1\}$, $H=\{0,1\}$