Difference between revisions of "Logical Equivalence"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
Line 1: Line 1:
−
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==

Revision as of 14:43, 14 October 2021

The equivalence of two statements 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 is the statement is that and have the same truth value. Another way of say this is that implies and implies .

Some ways to phrase this are

is equivalent to .
if and only if .
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} exactly when 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} .
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} iff 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 is an abbreviation for if and only if).
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} is a necessary and sufficient condition for 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} .

Examples:

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 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} iff 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} is True when 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 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} have the same truth values, and False when they have different truth values. In other words 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} iff 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} is True when 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 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} are both True or both False, 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 P} iff 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} is False is one of 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 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} is True while the other is false. In tabular form:

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} 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} 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 \iff Q}
True True True
True False False
False True False
False False True

The logical symbol for implication is "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 \iff} ", so you can write 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 \iff Q} for 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} iff 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} .

The statement

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} iff 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}

states that the implication

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 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}

and its converse are both true.


Resources

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Logical_Equivalence&oldid=2291"