If the statement "all cricketers are rich persons' is given as false which of the following Statement(s) remain(s) undetermined?

A. Some cricketers are not rich persons.

B. No cricketers are rich persons.

C. Some cricketers are rich persons.

Choose the correct answer from the options given below: 

Key Points

The correct answer is B and C only.

• The statement "all cricketers are rich persons" is a universal statement. Universal statements make a claim about all members of a group. In this case, the group is "cricketers" and the claim is that all cricketers are rich persons.
• A universal statement can be proven false by finding a single counterexample. In this case, finding a single cricketer who is not rich would prove the statement false. However, finding a single counterexample does not prove the statement true. There could be other cricketers who are rich, even if we find one who is not.
• Therefore, the statements "some cricketers are not rich persons" and "some cricketers are rich persons" remain undetermined.
• We cannot say for sure whether or not all cricketers are rich, or whether or not some cricketers are not rich.
• The statement "no cricketers are rich persons" is contradictory to the original statement.
• A contradictory statement makes the opposite claim of the original statement. In this case, the original statement claims that all cricketers are rich persons. The contradictory statement claims that no cricketers are rich persons.
• Since a contradictory statement is the opposite of the original statement, it is automatically false. Therefore, the statement "no cricketers are rich persons" is false.