Logical Equivalence

In mathematics, two statements are logically equivalent if they produce the same truth value in every case. For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \and Q } and $Q\land P$ are logically equivalent, as are $P\lor Q$ and $Q\lor P$, and $P\iff Q$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \implies Q } ) and its contrapositive (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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