State, whether or not the following sets are empty:{Prime numbers divisible by $ 2 $ }.

Hint: Check the set of prime numbers and then check that if there is any prime number that is divisible by $ 2 $ or a prime number that is even. Then check if the set is empty or not.

Complete step-by-step answer:
A prime number is a number which is divisible by only two numbers i.e. the number has only two factors \[1\] and the number itself.
Now take all the prime numbers and form a set of these numbers that is called the set of prime numbers.
The set of prime numbers is $ \left\{ {2,3,5,7,11 \ldots } \right\} $ and so on all the rest prime numbers are odd.
The set of prime numbers contains only one even prime number and rest all the prime numbers are odd.
The number $ 2 $ is a prime number and it is divisible by itself or we can say that $ 2 $ is divisible by $ 2 $ .
So, the set of prime numbers that is divisible by $ 2 $ is $ \left\{ 2 \right\} $ .
The set of prime numbers that are divisible by $ 2 $ contains one element.
So, the set of prime numbers is a singleton set not an empty set.
So, the correct answer is “The set of prime numbers is a singleton set not an empty set”.

Note: All the prime numbers are odd except the prime number $ 2 $ . The odd numbers are never divisible by $ 2 $ . Check the divisibility with the help of the divisibility rules. Therefore we get a single element 2 in the set which is also called a singleton set.