Tool By Zeal Point

Equal Sets : Extercise 1.2 Practice

Extercise 1.2

1. Which of the following are examples of the null set ?

(i) Set of odd natural numbers divisible by 2
Ans: Null set (∅)
Reason: No odd number is divisible by 2.

(ii) Set of even prime numbers
Ans: Not a null set
Reason: 2 is an even prime number.
So, the set is {2}.

(iii) {x : x is a natural number, x < 5 and x > 7}
Ans: Null set (∅)
Reason: No natural number can be less than 5 and greater than 7 at the same time.

(iv) {y : y is a point common to any two parallel lines}
Ans: Null set (∅)
Reason: Two parallel lines have no common point.

2. Which of the following sets are finite or infinite ?

(i) The set of months of a year
Ans: Finite set
Reason: There are 12 months.

(ii) {1, 2, 3, ...}
Ans: Infinite set
Reason: The counting numbers continue without an end.

(iii) {1, 2, 3, ..., 99, 100}
Ans: Finite set
Reason: It contains 100 elements.

(iv) The set of positive integers greater than 100
Ans: Infinite set
Reason: The numbers continue indefinitely: 101, 102, 103, ...

(v) The set of prime numbers less than 99
Ans: Finite set
Reason: There are only a limited number of prime numbers less than 99.

3. State whether each of the following sets is finite or infinite.

(i) The set of lines which are parallel to the x-axis
Ans: Infinite set
Reason: There are infinitely many lines parallel to the x-axis.

(ii) The set of letters in the English alphabet
Ans: Finite set
Reason: There are 26 letters.

(iii) The set of numbers which are multiples of 5
Ans: Infinite set
Reason: For example: 5, 10, 15, 20, 25, ... and so on.

(iv) The set of animals living on the Earth
Ans: Finite set
Reason: At any given time, the number of animals living on Earth is limited.

(v) The set of circles passing through the origin (0, 0)
Ans: Infinite set
Reason: Infinitely many different circles can pass through the same point (0, 0).

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4. In the following, state whether $A = B$ or not. Give reasons.

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

$Solution:$

The order of elements does not matter in a set.

Therefore,

$A = B$

Ans: Equal sets.

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

$Solution:$

Here, $12 \in A1$ but $12 \notin B$

Also, $18 \in B$ but $18 \notin A$

Therefore, $A \neq B$

Ans: Not equal sets.

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

$Solution:$

The positive even integers less than or equal to 10 are: $B = {2,4,6,8,10}$

Therefore, $A = B$

Ans: Equal sets.

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

$Solution:$

$A = {10, 20, 30, 40, 50,\dots}$

But,

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

Here, $15 \in B$ but $15 \notin A .$

Therefore, $A \neq B$

Ans: Not equal sets.

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Q5. Are the following pairs of sets equal ? Give reasons.

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

$Solution:$

$x 2 + 5x + 6 = 0$

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

Therefore,

$x = -2, -3$

So,

$B = {-2,-3}$

Since $A = {2, 3}$ and $B = {-2, -3}$

$A \neq B$

Ans: No, the sets are not equal.

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

$Solution:$

The letters in FOLLOW are:

A = {F, O, L, W}

(repeated letters are written only once)

Now,

$B = {y : y is a letter in the word WOLF}$

The letters in WOLF are: $B = {W, O, L, F}$

Since the order of elements does not matter in a set, $A = B$

Ans: Yes, the sets are equal.

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Q6. From the sets given below, select 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}

$Solution:$

Given: 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}

Compare the sets:

1. $B$ and $D$

$B = {1,2,3,4},$ $D = {3,1,4,2}$

They contain exactly the same elements. The order does not matter.

$B = D$

2. $E$ and $G$

$E = {-1, 1}$ $G = {1, -1}$

They contain exactly the same elements.

$E = G$

The other sets do not have exactly the same elements as any other set.

Ans: $B = D and E = G$

Note: Two sets are equal when they contain exactly the same elements, regardless of their order.

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