Beginning Activity Beginning Activity 2: Converse and Contrapositive
We now define two important conditional statements that are associated with a given conditional statement. If \(P\) and \(Q\) are statements, then The converse of the conditional statement \(P \to Q\) is the conditional statement \(Q \to P\text{.}\) The contrapositive of the conditional statement \(P \to Q\) is the conditional statement \(\mynot Q \to \mynot P\text{.}\)Definition.
1.
For the following, the variable \(x\) represents a real number. Label each of the following statements as true or false.
(a)
If \(x = 3\text{,}\) then \(x^2 = 9\text{.}\)
(b)
If \(x^2 = 9\text{,}\) then \(x = 3\text{.}\)
(c)
If \(x^2 \ne 9\text{,}\) then \(x \ne 3\text{.}\)
(d)
If \(x \ne 3\text{,}\) then \(x^2 \ne 9\text{.}\)
2.
Which statement in the list of conditional statements in ExerciseĀ 1 is the converse of TaskĀ 1.a? Which is the contrapositive of TaskĀ 1.a?
3.
Complete appropriate truth tables to show that
(a)
\(P \to Q\) is logically equivalent to its contrapositive \(\mynot Q \to \mynot P\text{.}\)
(b)
\(P \to Q\) is not logically equivalent to its converse \(Q \to P\text{.}\)