Q:

Write a conditional statement. Write the converse, inverse and contrapositive for your statement and determine the truth value of each. If a statements truth value is false, give a counterexample

Accepted Solution

A:
A conditional statement involves 2 propositions, p and q. The conditional statement, is a proposition which we write as: p⇒q,

and read  "if p then q"

Let p be the proposition: Triangle ABC is a right triangle with m(C)=90°.

Let q be the proposition: The sides of triangle ABC are such that

$$|AB|^2=|BC|^2+|AC|^2$$.

An example of a conditional statement is : p⇒q, that is:

if Triangle ABC is a right triangle with m(C)=90° then The sides of triangle ABC are such that $$|AB|^2=|BC|^2+|AC|^2$$

This compound proposition (compound because we formed it using 2 other propositions) is true. So the truth value is True,

the converse, inverse and contrapositive of p⇒q are defined as follows:

converse: q⇒p
inverse: ¬p⇒¬q (if [not p] then [not q])
contrapositive: ¬q⇒¬p

Converse of our statement:

if The sides of triangle ABC are such that $$|AB|^2=|BC|^2+|AC|^2$$
then Triangle ABC is a right triangle with m(C)=90°

True

Inverse of the statement:

if Triangle ABC is not a right triangle with m(C) not =90° then The sides of triangle ABC are not such that $$|AB|^2=|BC|^2+|AC|^2$$

True

Contrapositive statement:

if The sides of triangle ABC are not such that $$|AB|^2=|BC|^2+|AC|^2$$  then Triangle ABC is not a right triangle with m(C)=90°

True