Question

which statement is logically equivalent to the following conditional statement?

\if it has exactly four sides, then it is not a hexagon.\

• if it does not have exactly four sides, then it is not a hexagon.
• if it does not have exactly four sides, then it is a hexagon.
• if it is a hexagon, then it does not have exactly four sides.
• if it is not a hexagon, then it has exactly four sides.

Explanation:

Identify the conditional statement components

The given conditional statement is: "If it has exactly four sides, then it is not a hexagon."
Let \(p\) be "it has exactly four sides".
Let \(q\) be "it is not a hexagon".
The statement is represented symbolically as \(p
ightarrow q\).

Determine the logically equivalent form

A conditional statement \(p
ightarrow q\) is logically equivalent to its contrapositive, \(
eg q
ightarrow
eg p\).
The negation of \(q\) (\(
eg q\)) is: "it is a hexagon".
The negation of \(p\) (\(
eg p\)) is: "it does not have exactly four sides".
Therefore, the contrapositive is: "If it is a hexagon, then it does not have exactly four sides."

Match with the given options

Option 1: "If it does not have exactly four sides, then it is not a hexagon." (\(
eg p
ightarrow q\))
Option 2: "If it does not have exactly four sides, then it is a hexagon." (\(
eg p
ightarrow
eg q\))
Option 3: "If it is a hexagon, then it does not have exactly four sides." (\(
eg q
ightarrow
eg p\))
Option 4: "If it is not a hexagon, then it has exactly four sides." (\(q
ightarrow p\))

The third option matches the contrapositive.

Answer:

• (A) If it does not have exactly four sides, then it is not a hexagon.
• (B) If it does not have exactly four sides, then it is a hexagon.
• (C) If it is a hexagon, then it does not have exactly four sides. (Correct answer)
• (D) If it is not a hexagon, then it has exactly four sides.