Logical Equivalence - Revision history

Khanh at 02:32, 15 November 2021

2021-11-15T02:32:11Z

Lila at 20:45, 14 October 2021

2021-10-14T20:45:00Z

Lila at 20:43, 14 October 2021

2021-10-14T20:43:34Z

Lila at 18:02, 24 September 2021

2021-09-24T18:02:28Z

Lila: Created page with "In mathematics, two statements are logically equivalent if they produce the same truth value in every case. For example, "x is greater than 7" and "x is not less than or equal..."

2021-09-24T17:58:24Z

Created page with "In mathematics, two statements are logically equivalent if they produce the same truth value in every case. For example, "x is greater than 7" and "x is not less than or equal..."

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In mathematics, two statements are logically equivalent if they produce the same truth value in every case. For example, "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 (<math> P /implies Q </math>) and its contrapositive (<math> \neg Q /implies \neg P </math>) 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==
* [https://en.wikipedia.org/wiki/Logical_equivalence#:~:text=From%20Wikipedia%2C%20the%20free%20encyclopedia,truth%20value%20in%20every%20model. Logical Equivalence], Wikipedia
* [https://math.libretexts.org/Courses/Monroe_Community_College/MATH_220_Discrete_Math/2%3A_Logic/2.5%3A_Logical_Equivalences Logical Equivalences], Mathematics LibreTexts
* [https://www.usna.edu/Users/cs/roche/courses/f19sm242/get.php?f=slides2_1b.pdf Logical Equivalence and Truth Tables], United States Naval Academy College of Mathematics

← Older revision		Revision as of 20:43, 14 October 2021
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−	~~In mathematics,~~ two statements ~~are logically equivalent if they produce~~ the same truth value ~~in every case~~. ~~For example,~~ <math> P ~~\and~~ Q </math> and <math> Q ~~\and~~ P </math> are ~~logically~~ equivalent~~, as are~~ <math> P ~~\or Q~~ </math> and <math> Q ~~\or~~ P </math>~~, and~~ <math> P \iff Q </math> and <math> Q ~~\iff P~~ </math>. "~~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 (~~<math> P ~~\implies~~ Q </math>~~) and its contrapositive (~~<math> \~~neg~~ Q \~~implies~~ \~~neg~~ P </math>~~) 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"~~.	+	The equivalence of two statements <math>P</math> and <math>Q</math> is the statement is that <math>P</math> and <math>Q</math> have the same truth value. Another way of say this is that <math>P</math> implies <math>Q</math> and <math>Q</math> implies <math>P</math>.
		+
		+	Some ways to phrase this are
		+	:<math>P</math> is equivalent to <math>Q</math>.
		+	:<math>P</math> if and only if <math>Q</math>.
		+	:<math>P</math> exactly when <math>Q</math>.
		+	:<math>P</math> iff <math>Q</math>. (iff is an abbreviation for if and only if).
		+	:<math>P</math> is a necessary and sufficient condition for <math>Q</math>.
		+
		+	Examples:
		+	{\| class="wikitable" style="text-align: left"
		+	!First statement
		+	!Second statement
		+	!Equivalence
		+	\|-
		+	\| 4 is even.
		+	\| 6 is odd.
		+	\| 4 is even iff 6 is odd.
		+	\|-
		+	\| Triangle ABC is equilateral.
		+	\| Triangle ABC is equiangular.
		+	\| Triangle ABC is equilateral exactly when it is equiangular.
		+	\|}
		+
		+	The equivalence <math>P</math> iff <math>Q</math> is True when <math>P</math> and <math>Q</math> have the same truth values, and False when they have different truth values. In other words <math>P</math> iff <math>Q</math> is True when <math>P</math> and <math>Q</math> are both True or both False, and <math>P</math> iff <math>Q</math> is False is one of <math>P</math> and <math>Q</math> is True while the other is false. In tabular form:
		+	{\| class="wikitable" style="border=1 align=center"
		+	!width=50\|<math>P</math>
		+	!width=50\|<math>Q</math>
		+	!width=100\|<math>P \iff Q</math>
		+	\|---- align=center
		+	\|True
		+	\|True
		+	\|True
		+	\|---- align=center
		+	\|True
		+	\|False
		+	\|False
		+	\|---- align=center
		+	\|False
		+	\|True
		+	\|False
		+	\|---- align=center
		+	\|False
		+	\|False
		+	\|True
		+	\|}
		+
		+	The logical symbol for implication is "<math>\iff</math>",
		+	so you can write <math>P \iff Q</math> for <math>P</math> iff <math>Q</math>.
		+
		+	The statement
		+	:<math>P</math> iff <math>Q</math>
		+	states that the implication
		+	:<math>P</math> implies <math>Q</math>
		+	and its converse are both true.
		+

	==Resources==		==Resources==

← Older revision		Revision as of 18:02, 24 September 2021
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−	In mathematics, two statements are logically equivalent if they produce the same truth value in every case. For example, "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 (<math> P /implies Q </math>) and its contrapositive (<math> \neg Q /implies \neg P </math>) 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".	+	In mathematics, two statements are logically equivalent if they produce the same truth value in every case. For example, <math> P \and Q </math> and <math> Q \and P </math> are logically equivalent, as are <math> P \or Q </math> and <math> Q \or P </math>, and <math> P \iff Q </math> and <math> Q \iff P </math>. "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 (<math> P \implies Q </math>) and its contrapositive (<math> \neg Q \implies \neg P </math>) 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==		==Resources==

@@ Line 54: / Line 54: @@
 :<math>P</math> implies <math>Q</math>
 and its converse are both true.
 ==Resources==
 * [https://en.wikipedia.org/wiki/Logical_equivalence#:~:text=From%20Wikipedia%2C%20the%20free%20encyclopedia,truth%20value%20in%20every%20model. Logical Equivalence], Wikipedia
 * [https://math.libretexts.org/Courses/Monroe_Community_College/MATH_220_Discrete_Math/2%3A_Logic/2.5%3A_Logical_Equivalences Logical Equivalences], Mathematics LibreTexts
 * [https://www.usna.edu/Users/cs/roche/courses/f19sm242/get.php?f=slides2_1b.pdf Logical Equivalence and Truth Tables], United States Naval Academy College of Mathematics