Beginning Activity 2: Converse and Contrapositive

Beginning Activity Beginning Activity 2: Converse and Contrapositive
A4 US

We now define two important conditional statements that are associated with a given conditional statement.

Definition.

If $P$ and $Q$ are statements, then

The converse of the conditional statement $P \to Q$ is the conditional statement $Q \to P .$
The contrapositive of the conditional statement $P \to Q$ is the conditional statement $\neg Q \to \neg P .$

1.

For the following, the variable $x$ represents a real number. Label each of the following statements as true or false.

(a)

If $x = 3,$ then $x^{2} = 9 .$

(b)

If $x^{2} = 9,$ then $x = 3 .$

(c)

If $x^{2} \neq 9,$ then $x \neq 3 .$

(d)

If $x \neq 3,$ then $x^{2} \neq 9 .$

2.

Which statement in the list of conditional statements in Exercise 1 is the converse of Task 1.a? Which is the contrapositive of Task 1.a?

3.

Complete appropriate truth tables to show that

(a)

$P \to Q$ is logically equivalent to its contrapositive $\neg Q \to \neg P .$

(b)

$P \to Q$ is not logically equivalent to its converse $Q \to P .$

Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

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