Which statement is logically equivalent to the following unconditional statement? “If it has exactly four sides, then it is not a hexagon.”

Answer 1

To find a statement that is logically equivalent to the unconditional statement "If it has exactly four sides, then it is not a hexagon," we can rephrase it using the structure of logical equivalences.

The original statement can be represented as:

• Let \( p \) be the statement "It has exactly four sides."
• Let \( q \) be the statement "It is a hexagon."

The original statement can be expressed in logical form as \( p \rightarrow \neg q \) (If \( p \), then not \( q \)).

A logically equivalent form of an implication \( p \rightarrow \neg q \) can be expressed using contrapositive, which states that \( \neg q \rightarrow \neg p \) is logically equivalent.

Therefore, the contrapositive of the original statement is:

• "If it is a hexagon, then it does not have exactly four sides."

This statement is logically equivalent to the original statement. Hence, the equivalent statement to "If it has exactly four sides, then it is not a hexagon" is: "If it is a hexagon, then it does not have exactly four sides."