Logical Equivalence

In mathematics, two statements are logically equivalent if they produce the same truth value in every case. For example, $P\land Q$ and $Q\land P$ are logically equivalent, as are $P\lor Q$ and $Q\lor P$, and $P\iff Q$ and $Q\iff P$. "x is greater than 7" and "x is not less than or equal to 7" are logically equivalent because they are both true or both false simultaneously for every real number x. A conditional ($P\implies Q$) and its contrapositive ($\neg Q\implies \neg P$) are always logically equivalent. For example, "if x is even, then x is divisible by 2" is logically equivalent to its contrapositive, "if x is not divisible by 2, then x is not even".

Resources

• Logical Equivalence, Wikipedia
• Logical Equivalences, Mathematics LibreTexts
• Logical Equivalence and Truth Tables, United States Naval Academy College of Mathematics