Question

Which of the following sets are equal?
(i) $A = \\{ 1,2,3,4\\}$, $B = \\{ 4,3,2,1\\}$
(ii) $A = \\{ 4,8,12,16\\}$, $B = \\{ 8,4,16,18\\}$
(iii) $X = \\{ 2,4,6,8\\}$ ,$Y = \ \\{\ x:x\ \text{is a positive even integer} \ 0 < x < 10\ \\}$
(iv) $P = \ \\{\ x:x\ \text{is a multiple of}\ 10,\ x \in N\ \\}$ , $Q = \\{ 10,15,20,25,30,\ldots\\}$

Answer 1

In this question, we need to find which of the following given sets are equal. Two sets are said to be equal if they have all the elements the same. Even if the order of the elements is not the same, elements should be the same. In each option, we have given a pair of sets and some sets are given in roaster form and some in set builder form. So, first we will find all the elements in each set by writing them in roaster form and check whether they are equal or not.

Complete step by step answer:
Let’s start the procedure of checking options step by step.
(i) $A = \\{ 1,2,3,4\\}$, $B = \\{ 4,3,2,1\\}$
Now on rewriting the elements in set B ,
We get,
$A = \\{ 1,2,3,4\\}$ and $B = \\{ 1,2,3,4\\}$
Now on observing both the sets all the elements in both the sets are equal and same.
Thus $A = B$
Therefore we can conclude that the set $A = \\{ 1,2,3,4\\}$ is equal to the set $B = \\{ 4,3,2,1\\}$ .

(ii) $A = \\{ 4,8,12,16\\}$, $B = \\{ 8,4,16,18\\}$
Now on observing all the elements of both the set, the element $12$ belongs to set $A$ which doesn’t belong to set $B$ and also the element $18$ belongs to set $B$ which doesn’t belong to set $A$.
Thus $A \neq B$
Therefore we can conclude that the set $A = \\{ 4,8,12,16\\}$ is not equal to the set $B = \\{ 8,4,16,18\\}$.

(iii) $X = \\{ 2,4,6,8\\}$ , $Y = \ \\{\ x:x\ \text{is a positive even integer} \ 0 < x < 10\ \\}$
First let us convert the set $Y$ to roster form.
Given that
$Y = \ \\{\ x:x\ \text{is a positive even integer} \ 0 < x < 10\ \\}$
So the possible values of $x$ are $2,4,6,8$
Therefore $Y = \\{ 2,4,6,8\\}$
Now on observing both the sets all the elements in both the sets are equal and same.
Thus $X = Y$
Therefore we can conclude that the set $X = \\{ 2,4,6,8\\}$ is equal to the set $Y = \ \\{\ x:x\ \text{is a positive even integer} \ 0 < x < 10\ \\}$ .

(iv) $P = \ \\{\ x:x\ \text{is a multiple of}\ 10,\ x \in N\ \\}$ , $Q = \\{ 10,15,20,25,30,\ldots\\}$
First let us convert the set P to roster form.
Given that
$P = \ \\{\ x:x\ \text{is a multiple of}\ 10,\ x \in N\ \\}$
So the possible values of $x$ are $10,20,30,40,\ldots$
Therefore $P = \\{ 10,20,30,40,\ldots\\}$
On observing both the sets most of the elements in set $P$ don't belong to set $Q$.
Thus $P \neq Q\$
Therefore we can conclude that set $P = \ \\{\ x:x\ \text{is a multiple of}\ 10,\ x \in N\ \\}$ is not equal to set $Q = \\{ 10,15,20,25,30,\ldots\\}$

Final answer :
(i). The set $A = \\{ 1,2,3,4\\}$ is equal to the set $B = \\{ 4,3,2,1\\}$ .
(ii). The set $A = \\{ 4,8,12,16\\}$ is not equal to the set $B = \\{ 8,4,16,18\\}$.
(iii). The set $X = \\{ 2,4,6,8\\}$ is equal to the set $Y = \ \\{\ x:x\ \text{is a positive even integer} \ 0 < x < 10\ \\}$ .
(iv). The set $P = \ \\{\ x:x\ \text{is a multiple of}\ 10,\ x \in N\ \\}$ is not equal to set $Q = \\{ 10,15,20,25,30,\ldots\\}$

Note:

We need to remember that two sets are said to be equal when the number of elements is equal and all the elements are the same. We also need to be aware that equal sets and equivalent sets are two different things.
Two sets are said to be equivalent sets, when the number of terms in each set are equal, also there is no condition that the elements must be the same.
We can say that all equal sets are equivalent sets but there is no guarantee that all equivalent sets can be equal sets.