Question

State whether the set of even prime numbers is empty or not?

Answer 1

We know that a set is said to be empty set if the set does not have any elements. The size (or) cardinality is equal to zero in an empty set. We know that a prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A prime number has only 1 and the number itself as the factors. Now we should find the set of prime numbers. Then we should check the cardinality value of the set of even primes. Now we should check whether the condition is satisfied or not.

Complete step-by-step solution:
Before solving the question, we should know the definition of an empty set. A set is said to be empty set if the set does not have any elements. The size (or) cardinality is equal to zero in an empty set.
From the question, we should find whether a set of prime numbers is an empty set or not.
We know that a prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A prime number has only 1 and the number itself as the factors.
We know that 2 is the only even prime number. Because 2 is greater than 1 and 2 is having only 2 and 1 as factors. So, we can say that 2 is a prime number.
Hence, we can say that the set of even prime numbers is equal to \left\\{ 2 \right\\}.
Hence, we can say that the number of elements in the set of prime numbers is equal to 1.
So, the cardinality value of the set of prime numbers is equal to 1.
But we know that a set is said to be empty set if the set does not have any elements. The size (or) cardinality is equal to zero in an empty set.
Hence, we can say that the set of prime numbers is not an empty set.

Note: Students may have a misconception that a set which is said to have only one element is said to be an empty set. They may have a misconception that the cardinality of the set is equal to 1. Then we can say that the set of even prime numbers is an empty set. But we know that the set of prime numbers is not an empty set. So, this misconception should be avoided. So, students should remember that a set which is said to have only one element is said to be a singleton set.