Question

A certain conditional is false. Must its converse be false? Must its inverse be false? Must its contrapositive be false?

Short answer

If the conditional statement is false, the contrapositive, inverse, and converse statements also false.

Step-by-step solution

Step 1. Apply the concept of conditional statement and contrapositive.

Conditional statements are thosestatements where a hypothesis is followed by a conclusion. It is also known as an " If-then" statement.

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.

Step 2. Apply the concept of inverse and converse of the conditional statement.

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.

A converse statement is obtained by exchanging the positions of 'p' and 'q' in the condition.

Step 3. Step description.

Consider the conditional statement “If two lines are coplanar, then they do not intersect” and this is a false statement.

Write the contrapositive, inverse and converse of the conditional statement as follows:

Conditional: If two lines are coplanar, then they do not intersect. false statement.

Converse: If two lines do not intersect, then they are coplanar. False statement.

Inverse: If two lines are not coplanar, then they intersect. False statement.

Contra positive: If two lines intersect, then they are not coplanar. False statement.

Therefore, if the conditional statement is false, then contrapositive, inverse and converse statements also false.