Question

When proving theorems, mathematicians consider most frequently four statements:

1. the conditional statement

2. the converse to the conditional statement

3. the inverse

4. the contrapositive

Among those four statements, which ones require negation of parts, or all, of the conditional statement? When negating a statement, for which keywords must you search? How are they negated?

Answer 1

A conditional statement is an if then type statement which consists of a hypothesis and a conclusion. For example: If X is an odd number then X is a Prime Number. In this Statement the "if" part X is an odd number is the Hypothesis and the "then" part X is a prime number is the conclusion. This statement is false as 1 is odd but not a prime number.

The converse of a conditional statement simply reverse the role of hypothesis and conclusion. For Example: the converse of above statement is " If X is a Prime Number Then X is an odd Number.The converse the conditional statement does not require negation of any of the part of the statement.

The inverse of a conditional statement Negates both the Hypothesis part and the conclusion part of the statement. For Example: For the above conditional statement, its inverse is: "If X is Not an odd number then X is Not an Prime number. This statement is false as 2 is not an odd number but 2 is a prime number.The inverse of the conditional statement can have different truth value than of the statement itself.

The contrapositive of a a conditional statement is logically equivalent to the statement itself, i.e. both has the same truth value. the contrapositive can be obtained by negating both if and then part and switching the role of hypothesis and conclusion. For example: For the above statement its contrapositive is " If X is not a prime number Then X is not an odd number.