Question

Write the converse, inverse and contrapositive of the following statements:
“If a function is differentiable then it is continuous”.

Answer 1

Hint:First of all, divide the conditional statement into hypothesis $p$ and conclusion $q$ of the statement. Then find the converse by if $q$, then $p$; the inverse by if not $p$, then not $q$ and the contrapositive by if not $q$, then not $p$. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:
Let the statement “The function is differentiable” be $p$.
And the statement “The function is continuous” be $q$.
The given conditional statement i.e., “If a function is differentiable then it is continuous” is if-then statement which can be written as $p \to q$.
To form the converse of the conditional statement, interchange the hypothesis and the conclusion which is given by $q \to p$.
So, the converse of the given statement is “If a function is continuous then it is differentiable”.
To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion which is given by $\sim p \to \sim q$.
So, the inverse of the given statement is “If a function is not differentiable then it is not continuous”.
To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement which is given by $\sim q \to \sim p$.
So, the contrapositive of the given statement is “If a function is not continuous then it is not differentiable”.

Note:A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. Here $p$ is the hypothesis and $q$ is the conclusion of the given conditional. statement.