Question

suppose that the given statements are true.
find the other true statements.
(a) true statement: if deon prefers black to red, then i liked the poem.
true statement: if i liked the poem, then the passports are missing.
which statement must also be true?
select
(b) true statement: if austins cousin is a girl, then the window is above the door.
true statement: austins cousin is a girl.
which statement must also be true?
select
(c) true statement: if our instructor hates to fly, then the wall has been painted.
which statement must also be true?
select

Explanation:

Step1: Apply hypothetical syllogism (a)

If \(P
ightarrow Q\) and \(Q
ightarrow R\), then \(P
ightarrow R\).
Let \(P=\text{Deon prefers black to red}\), \(Q=\text{I liked the poem}\), \(R=\text{passports are missing}\).
So \(P
ightarrow R\): If Deon prefers black to red, then the passports are missing.

Step2: Apply modus ponens (b)

If \(P
ightarrow Q\) and \(P\) is true, then \(Q\) is true.
Let \(P=\text{Austin's cousin is a girl}\), \(Q=\text{window is above the door}\).
Since \(P\) is true, \(Q\) must be true: The window is above the door.

Step3: Identify contrapositive (c)

The contrapositive of \(P
ightarrow Q\) is \(
eg Q
ightarrow
eg P\), which is logically equivalent.
Let \(P=\text{our instructor hates to fly}\), \(Q=\text{the wall has been painted}\).
Contrapositive: If the wall has not been painted, then our instructor does not hate to fly.

Answer:

(a) If Deon prefers black to red, then the passports are missing.
(b) The window is above the door.
(c) If the wall has not been painted, then our instructor does not hate to fly.