Question

the conditional statement ( p
ightarrow q )
the converse ( q
ightarrow p )
the inverse ( sim p
ightarrow sim q )
the contrapositive ( sim q
ightarrow sim p )
:: if it is not valentine day, then it is february.
:: if it is not february, then it is not valentine’s day.
:: if it is valentines day, then it is not february.
:: if it is not valentine’s day, then it is not february.
:: if it is valentines day, then it is february.
:: if it is february, then it is valentine’s day.

Explanation:

To solve this, we analyze each logical form:

1. Conditional (\( p \to q \))

Let \( p \): "It is Valentine’s Day", \( q \): "It is February".
\( p \to q \) means If \( p \), then \( q \).
Match: If it is Valentine’s Day, then it is February.

2. Converse (\( q \to p \))

The converse swaps \( p \) and \( q \), so \( q \to p \) means If \( q \), then \( p \).
Match: If it is February, then it is Valentine’s Day.

3. Inverse (\( \sim p \to \sim q \))

The inverse negates both \( p \) and \( q \), so \( \sim p \to \sim q \) means If not \( p \), then not \( q \).
Match: If it is not Valentine’s Day, then it is not February.

4. Contrapositive (\( \sim q \to \sim p \))

The contrapositive negates and swaps \( p \) and \( q \), so \( \sim q \to \sim p \) means If not \( q \), then not \( p \).
Match: If it is not February, then it is not Valentine’s Day.

Final Matching:

• The conditional \( p \to q \): If it is Valentine’s Day, then it is February.
• The converse \( q \to p \): If it is February, then it is Valentine’s Day.
• The inverse \( \sim p \to \sim q \): If it is not Valentine’s Day, then it is not February.
• The contrapositive \( \sim q \to \sim p \): If it is not February, then it is not Valentine’s Day.

(Note: The other options are incorrect. For example, "If it is not Valentine day, then it is February" and "If it is Valentine’s day, then it is not February" do not follow the logical forms.)

Answer:

To solve this, we analyze each logical form:

1. Conditional (\( p \to q \))

Let \( p \): "It is Valentine’s Day", \( q \): "It is February".
\( p \to q \) means If \( p \), then \( q \).
Match: If it is Valentine’s Day, then it is February.

2. Converse (\( q \to p \))

The converse swaps \( p \) and \( q \), so \( q \to p \) means If \( q \), then \( p \).
Match: If it is February, then it is Valentine’s Day.

3. Inverse (\( \sim p \to \sim q \))

The inverse negates both \( p \) and \( q \), so \( \sim p \to \sim q \) means If not \( p \), then not \( q \).
Match: If it is not Valentine’s Day, then it is not February.

4. Contrapositive (\( \sim q \to \sim p \))

The contrapositive negates and swaps \( p \) and \( q \), so \( \sim q \to \sim p \) means If not \( q \), then not \( p \).
Match: If it is not February, then it is not Valentine’s Day.

Final Matching:

• The conditional \( p \to q \): If it is Valentine’s Day, then it is February.
• The converse \( q \to p \): If it is February, then it is Valentine’s Day.
• The inverse \( \sim p \to \sim q \): If it is not Valentine’s Day, then it is not February.
• The contrapositive \( \sim q \to \sim p \): If it is not February, then it is not Valentine’s Day.

(Note: The other options are incorrect. For example, "If it is not Valentine day, then it is February" and "If it is Valentine’s day, then it is not February" do not follow the logical forms.)