Exercise 7 Page 589

Practice makes perfect

We want to classify related conditional statement based on a conditional statement. Let's summarize how they are defined.

	Words	Symbols
Conditional statement	If p, then q	p → q
Converse	If q, then p	q → p
Inverse	If not p, then not q	~ q → ~ p
Contrapositive	If not q, then not p	~ p → ~ q
Biconditional statement	p if and only if q	p ↔ q

A conditional statement has two parts: a hypothesis and a conclusion. Let's identify these parts in the statement we have been given.

If you are a soccer player, then you are an athlete.

Let's write this using symbols, with the hypothesis as p and the conclusion as q. p → q We will now identify the parts in the related conditional statement.

If you are not a soccer player, then you are not an athlete.

We can also write this sentence using symbols. Recall that we write not using ~. ~ p → ~ q Both the hypothesis and the conclusion are negated. We recognize this as a contrapositive.

Let's look at the next related conditional statement.

If you are an athlete, then you are a soccer player.

Let's write this using symbols. q → p We see that the hypothesis and the conclusion have switched places. That means it is a converse.

We are now ready to study the third related conditional statement.

You are a soccer player, if and only if you are an athlete.

Let's write the statement using symbols, as this can help us classify the conditional statement. p ↔ q This kind of statement we recognize as a biconditional statement.

Let's look at the fourth conditional statement.

If you are not an athlete, then you are not a soccer player.

Since writing the statements using symbols can make the classification of the statement easier, we will do that now. ~ q → ~ p When both the hypothesis and the conclusion are negated and have switched places we have a contrapositive.

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Exercises p.588-589

Exercise 7 Page 589

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