Question

use the equivalency

\\(p \
ightarrow q \equiv \sim q \
ightarrow \sim p\\)

to select the statement that is logically equivalent to the conditional statement.

\if the light is on, then someone is home.\

• if someone is home, the light is on.
• the light is on and someone is home
• someone is not home, or the light is on.
• if someone is not home, the light is not on.

Explanation:

Define variables for the statements

$$
LATEXBLOCK0
$$

Apply the contrapositive equivalence

$$
LATEXBLOCK1
$$

Formulate the equivalent statement

$$ \text{"If someone is not home, then the light is not on."} $$

Answer:

• If someone is home, the light is on.
• The light is on and someone is home
• Someone is not home, or the light is on.
• If someone is not home, the light is not on. (Correct answer)