Question

State whether the following pairs of sets are equal or not:
A=\left\\{ 2,4,6,8,10 \right\\} , B=\left\\{ \text{even natural numbers} \right\\}

Answer 1

In this problem we need to check whether the given two sets are equal or not. We have given the values in one set. In another set we have given a statement that means the values in the second set should satisfy the given statement. We have given the statement for the second set as ‘even natural numbers’. We know that the natural numbers range from $0$ to $+\infty$ . Now we will list the even numbers that are divisible by $2$ as the values of the second set. After having the values in the second set we will compare both the sets to check whether they are equal or not.

Complete step by step solution:
Given sets are A=\left\\{ 2,4,6,8,10 \right\\} , B=\left\\{ \text{even natural numbers} \right\\} .
We can observe that in the first set we have given the values and in the second set we have given a statement.
To find the values in the second set we are going to consider the statement in the second set which is ‘even natural numbers’.
In this statement we can observe two different types of numbers which are ‘natural numbers’ and ‘even numbers’.
We know that the numbers which are having positive signs are called ‘natural numbers’ including $0$. So the natural numbers vary from $0$ to $+\infty$.
We know that the numbers which are divisible by $2$ and give $0$ as a reminder are called ‘even numbers’. So the even numbers are \left\\{ -\infty ,.....,-8,-6,-4,-2,0,2,4,6,8,......,+\infty \right\\} .
Now the even natural numbers means the numbers which are having positive signs and are divisible by $2$, 0 is not a natural number so we are not considering this in the set B . So the examples for even natural numbers are \left\\{ 2,4,6,8,10,12,......,+\infty \right\\} .
So the values which are supposed to be in the set $B$ are \left\\{2,4,6,8,10,12,......,+\infty \right\\}.
\begin{aligned} & B=\left\\{ \text{even natural numbers} \right\\} \\\ & \Rightarrow B=\left\\{2,4,6,8,10,12,......,+\infty \right\\} \\\ \end{aligned}
Observing the two sets we can say that all the values in set $B$ are not present in the set $A$ like $12$ , $14$ .
So that the two sets are not equal.
$\therefore A\ne B$ .

Note: In this problem we have only asked to check whether both the sets are equal or not. In some cases they may ask to check whether the set $A$ is a subset of set $B$ or not. We can observe that all the values in set $A$ are present in the set $B$. So we can say that the set $A$ is a subset of set $B$.