Question

Which one of the following is a finite set?
(a) $\left\{ x:x\in Z,x < 5 \right\}$
(b) $\left\{ x:x\in W,x\ge 5 \right\}$
(c) $\left\{ x:x\in N,x > 10 \right\}$
(d) $\left\{ x:x\text{ is an even prime number} \right\}$

Answer 1

Hint: We start solving the problem, by recalling the definitions of integers, whole numbers, natural numbers and prime numbers. We then write the numbers that were satisfying the given condition for x in each option. We then check whether the total number of terms present are finite or infinite to get the required answer.

Complete step-by-step answer:
According to the problem, we are asked to find which of the given options represent a finite set.
Let us first check option (a).
We have given the set $\left\{ x:x\in Z,x < 5 \right\}$. We know that $Z$ represents all the integers present on the number line. We can see that the set $\left\{ x:x\in Z,x < 5 \right\}$ represents integers that were less than 5.
Now, let us write all the integers present in the set $\left\{ x:x\in Z,x < 5 \right\}$.
So, we have $x=\left\{ -\infty ,......,-2,-1,0,1,2,3,4 \right\}$. We can see that there are infinite terms present in the set $\left\{ x:x\in Z,x < 5 \right\}$, which makes it an infinite set.
Let us first check option (b).
We have given the set $\left\{ x:x\in W,x\ge 5 \right\}$. We know that $W$ represents all the integers that were greater than or equal to 0 on the number line (Whole numbers). We can see that the set $\left\{ x:x\in W,x\ge 5 \right\}$ represents integers that were greater than or equal 5.
Now, let us write all the integers present in the set $\left\{ x:x\in W,x\ge 5 \right\}$.
So, we have $x=\left\{ 5,6,7,8,9,.........,\infty \right\}$. We can see that there are infinite terms present in the set $\left\{ x:x\in W,x\ge 5 \right\}$, which makes it an infinite set.
Let us first check option (c).
We have given the set $\left\{ x:x\in N,x > 10 \right\}$. We know that $N$ represents all the integers that were greater than 0 on the number line (Natural numbers). We can see that the set $\left\{ x:x\in N,x > 10 \right\}$ represents integers that were greater than 10.
Now, let us write all the integers present in the set $\left\{ x:x\in N,x > 10 \right\}$.
So, we have $x=\left\{ 10,11,12,13,14,.........,\infty \right\}$. We can see that there are infinite terms present in the set $\left\{ x:x\in W,x\ge 5 \right\}$, which makes it an infinite set.
Let us check option (d).
We have given the $\left\{ x:x\text{ is an even prime number} \right\}$. We know that the prime numbers are positive integers which is divided by 1 and itself. We know that all positive even numbers by 2. But 2 has factors 1 and itself. So, 2 is only even prime number.
So, we have $x=\left\{ 2 \right\}$ which makes it a finite set.
∴ The correct option for the given problem is (d).

So, the correct answer is “Option (d)”.

Note: Whenever we get this type of problem, we first write all the terms present in the sets by following the required standard definitions. We should know that the set is finite if we are able to count the total number of terms present in the set. Similarly, we can expect problems to check whether the set $\left\{ x:x\text{ is the number that represents population in the world} \right\}$ is finite or infinite.