Question

use the equivalency

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

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

\if you save half your paycheck, then you can buy a car.\

• if you do not save half your paycheck, then you cannot buy a car.
• you saved half your paycheck and you can buy a car.
• you do not save half your paycheck or you can buy a car.
• you do not save half your paycheck and you can buy a car.

Explanation:

Define variables for the component statements

\(p\): "You save half your paycheck."
\(q\): "You can buy a car."

Apply the conditional equivalency

$$ p ightarrow q \equiv \sim p \lor q $$

\(\sim p\): "You do not save half your paycheck."
\(q\): "You can buy a car."
\(\sim p \lor q\): "You do not save half your paycheck or you can buy a car."

Match with the given options

Option 1: "If you do not save half your paycheck, then you cannot buy a car." (\(\sim p
ightarrow \sim q\))
Option 2: "You saved half your paycheck and you can buy a car." (\(p \land q\))
Option 3: "You do not save half your paycheck or you can buy a car." (\(\sim p \lor q\))
Option 4: "You do not save half your paycheck and you can buy a car." (\(\sim p \land q\))

Answer:

• If you do not save half your paycheck, then you cannot buy a car.
• You saved half your paycheck and you can buy a car.
• You do not save half your paycheck or you can buy a car. (Correct answer)
• You do not save half your paycheck and you can buy a car.