Difference between revisions of "Quantifiers"
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Quantifiers are words, expressions, or phrases that indicate the number of elements that a statement pertains to. In mathematical logic, there are two quantifiers: the universal quantifier ("for all") and the existential quantifier ("there exists"). | Quantifiers are words, expressions, or phrases that indicate the number of elements that a statement pertains to. In mathematical logic, there are two quantifiers: the universal quantifier ("for all") and the existential quantifier ("there exists"). | ||
| − | Universal quantifier: This quantifier is used to state a proposition that is true for all variables x of a given set. For example, the proposition "x^2 is a nonnegative number" is true for | + | Universal quantifier: This quantifier is used to state a proposition that is true for all variables x of a given set. For example, the proposition "x^2 is a nonnegative number" is true for every real number, so we state "for all real numbers <math> x </math>, <math> x^2 </math> is nonnegative". Symbolically, we write the universal quantifier as <math> \forall </math> (for example, <math> \forall x \in \R, x^2 \ge 0</math>). |
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| + | Universal quantifier: This quantifier is used to state a proposition that is at least true for one element x of a given set. For example, the proposition "n is greater than 7" is not true for all real numbers, but we know it is true for some. | ||
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==Resources== | ==Resources== | ||
* [https://math.libretexts.org/Courses/Monroe_Community_College/MATH_220_Discrete_Math/2%3A_Logic/2.7%3A_Quantiers Quantifiers], Mathematics LibreTexts | * [https://math.libretexts.org/Courses/Monroe_Community_College/MATH_220_Discrete_Math/2%3A_Logic/2.7%3A_Quantiers Quantifiers], Mathematics LibreTexts | ||
Revision as of 20:33, 23 September 2021
Quantifiers are words, expressions, or phrases that indicate the number of elements that a statement pertains to. In mathematical logic, there are two quantifiers: the universal quantifier ("for all") and the existential quantifier ("there exists").
Universal quantifier: This quantifier is used to state a proposition that is true for all variables x of a given set. For example, the proposition "x^2 is a nonnegative number" is true for every real number, so we state "for all real numbers 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 } , is nonnegative". Symbolically, we write the universal quantifier 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 \forall } (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 \forall x \in \R, x^2 \ge 0} ).
Universal quantifier: This quantifier is used to state a proposition that is at least true for one element x of a given set. For example, the proposition "n is greater than 7" is not true for all real numbers, but we know it is true for some.
Resources
- Quantifiers, Mathematics LibreTexts
