Difference between revisions of "Logical Implication"
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| − | A logical implication is a relationship between two statements. If a statement Q is always true when another statement P is true, then we say that | + | A logical implication is a relationship between two statements. If a statement <math> Q </math> is always true when another statement <math> P </math> is true, then we say that <math> P </math> implies <math> Q </math>, which is denoted symbolically as <math> P \implies Q </math>. Note that if <math> P </math> is false, <math> Q </math> does not necessarily have to be false. For example, if <math> x > 10 </math>, then <math> x > 0 </math>, so we can say that "<math> x > 10 \implies x > 0 </math>". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, <math> x > 10 \implies x > 0 </math> does NOT mean that <math> x \le 10 \implies x \le 0 </math>. The truth table for logical implication is as follows: |
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| − | Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) does not necessarily have the same truth value as <math> P \implies Q </math>, the contrapositive (<math> \neg Q \implies \neg P </math>) does. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are logically equivalent, and always have the same truth value for any number x. | + | Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) does not necessarily have the same truth value as <math> P \implies Q </math>, the contrapositive (<math> \neg Q \implies \neg P </math>) does. For example, <math> x > 10 \implies x > 0 </math> and its contrapositive, <math> x \leq 0 \implies x \leq 10 </math>, are logically equivalent, and always have the same truth value for any number <math> x </math>. |
==Resources== | ==Resources== | ||
* [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html Truth Tables, Tautologies, and Logical Equivalences], Millersville University | * [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables.html Truth Tables, Tautologies, and Logical Equivalences], Millersville University | ||
Revision as of 13:13, 27 September 2021
A logical implication is a relationship between two statements. If a 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 Q } is always true when another statement is true, then we say that implies , which is denoted symbolically as . Note that if is false, does not necessarily have to be false. For example, 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 x > 10 } , then 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 x > 0 } , so we can say that "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 x > 10 \implies x > 0 } ". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That 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 x > 10 \implies x > 0 } does NOT mean that 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 x \le 10 \implies x \le 0 } . The truth table for logical implication is as follows:
| 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 \Rightarrow Q} |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Note that while the inverse 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 \implies Q } (that 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 \neg P \implies \neg Q } ) does not necessarily have the same truth value as 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 } , the 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 } ) does. 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 x > 10 \implies x > 0 } 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 x \leq 0 \implies x \leq 10 } , are logically equivalent, and always have the same truth value for any number 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 x } .
Resources
- Truth Tables, Tautologies, and Logical Equivalences, Millersville University
