Question

State the converse and contrapositive of each of the following statements:
(i) a : A positive integer is prime only if it has no divisors other than 1 and itself.
(ii) b : I go to a beach whenever it is a sunny day.
(iii) c : If it is hot outside then you feel thirsty.

Answer 1

Hint: We will first write the definitions of converse and contrapositive and then, write the given statements in the form required. After then apply all the definitions to given statements. Use the conditional statements topic.

Complete step-by-step answer:
Let us first write the definitions of converse and contrapositive:-
Converse:- If we have a statement with format “If p, then q” where p and q are the lines explaining the event, then its converse will be totally opposite that is “If q, then p”.
Contrapositive:- If we have a statement with format “If p, then q” where p and q are the lines explaining the event, then its contrapositive will be “If not q, then not p”. We can also represent this as “If $ \sim q$ then, $ \sim p$.
Let us now go to the first statement:
a : A positive integer is prime only if it has no divisors other than 1 and itself.
We can rewrite it as:
a: If a positive integer is prime, then it has no divisors other than 1 and itself.
Comparing it with “If p, then q”:
Here, p = a positive integer is prime and q = it has no divisors other than 1 and itself.
The converse of the statement will be “If q, then p” that is as follows:
If a positive integer has no divisors other than 1 and itself then it is prime.
The contrapositive of the statement will be “If $ \sim q$ then, $ \sim p$” that is as follows:
If a positive integer has divisors other than 1 and itself it is not prime.
Let us now go to the second statement:
b : I go to a beach whenever it is a sunny day.
We can rewrite it as:
b: If it is a sunny day then I go to a beach.
Comparing it with “If p, then q”:
Here, p = a it is a sunny day and q = I go to a beach.
The converse of the statement will be “If q, then p” that is as follows:
If I go to a beach then it is a sunny day.
The contrapositive of the statement will be “If $ \sim q$ then, $ \sim p$” that is as follows:
If I do not go to a beach then it is not a sunny day.
Let us now go to the third statement:
c : If it is hot outside then you feel thirsty.
Comparing it with “If p, then q”:
Here, p = it is hot outside and q = you feel thirsty.
The converse of the statement will be “If q, then p” that is as follows:
If you feel thirsty then it is hot outside.
The contrapositive of the statement will be “If $ \sim q$ then, $ \sim p$” that is as follows:
If you do not feel thirsty then it is not hot outside.

Note: We must not start with the given sentence always. We need to rewrite in the form “If p, then q”, because otherwise we may misplace p and q with each other.
Do not always tend to change the sentence because as in the statement c, we already had our desired form. SO, we didn’t change it.
Fact:- Contrapositive is always true if the original statement was true but converse need not be true. So, do not mistake all of these statements to be equivalent.