Difference between revisions of "Logical Implication"
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| − | Note that while the inverse of <math> P \implies Q </math> (that is, <math> \neg P \implies \neg Q </math>) | + | 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 both true or both false at the same time for all values of x. |
==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 13:52, 24 September 2021
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 "P implies Q", which is denoted symbolically 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 } . Note that if P is false, Q does not necessarily have to be false. For example, if x > 10, then x is also greater than 0, so we can say that "". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, does NOT mean that . The truth table for logical implication is as follows:
| 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 both true or both false at the same time for all values of x.
Resources
- Truth Tables, Tautologies, and Logical Equivalences, Millersville University
