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In mathematical logic, a sentence (or closed formula) [1] of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition , something that must be true or false.
The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a ...
First expand the domain from the set of all real numbers to a two-sorted domain, with the second sort containing all sets of real numbers. Add a new binary predicate to the language: the membership relation. Then sentences that were second-order become first-order, with the formerly second-order quantifiers ranging over the second sort instead.
In the latter case, a (declarative) sentence is just one way of expressing an underlying statement. A statement is what a sentence means, it is the notion or idea that a sentence expresses, i.e., what it represents. For example, it could be said that "2 + 2 = 4" and "two plus two equals four" are two different sentences expressing the same ...
A valid number sentence that is true: 83 + 19 = 102. A valid number sentence that is false: 1 + 1 = 3. A valid number sentence using a 'less than' symbol: 3 + 6 < 10. A valid number sentence using a 'more than' symbol: 3 + 9 > 11. An example from a lesson plan: [6] Some students will use a direct computational approach.
The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, [1] and the LaTeX symbol.
The predicate calculus goes a step further than the propositional calculus to an "analysis of the inner structure of propositions" [4] It breaks a simple sentence down into two parts (i) its subject (the object (singular or plural) of discourse) and (ii) a predicate (a verb or possibly verb-clause that asserts a quality or attribute of the object(s)).