ISI2018-DCG-5

Question

Let $A$ be the set of all prime numbers, $B$ be the set of all even prime numbers, and $C$ be the set of all odd prime numbers. Consider the following three statements in this regard:

1. $A=B\cup C$.
2. $B$ is a singleton set
3. $A = C \cup \{2\}$

Then which one of the following holds?

• A. None of the above statements is true.
• B. Exactly one of the above statements is true.
• C. Exactly two of the above statements are true.
• D. All the above three statements are true.

Answer 1

A={2,3,5,7,11,13,17,19........}

B={2}    ($\because$ 2 is the only even prime.Because,if any other prime is divisible by 2, then it is not prime)

C={3,5,7,11,13,17,19,23....} ($\because$all primes except 2 are odd.)

1-True.It can be clearly seen that $A= B \cup C$

2-True.Since,B has only 1 element.

3-True .$\because$ C need only one element 2 to become A.

$\therefore$all the statements are true.

Hence option D