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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.
In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of p {\displaystyle p} and q {\displaystyle q} is sometimes expressed as p ≡ q {\displaystyle p\equiv q} , p :: q {\displaystyle p::q} , E p q {\displaystyle {\textsf {E}}pq} , or p q ...
The closely related code point U+2262 ≢ NOT IDENTICAL TO (≢, ≢) is the same symbol with a slash through it, indicating the negation of its mathematical meaning. [ 1 ] In LaTeX mathematical formulas, the code \equiv produces the triple bar symbol and \not\equiv produces the negated triple bar symbol ≢ {\displaystyle \not ...
Structures over a signature, also known as models, provide formal semantics to a signature and the first-order language over it.. A structure over a signature consists of a set (known as the domain of discourse) together with interpretations of the non-logical symbols: Every constant symbol is interpreted by an element of and the interpretation of an -ary function symbol is an -ary function on ...
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 ...
Thus, the statement "If I am a triangle, then I am a three-sided polygon" is logically equivalent to "If I am a three-sided polygon, then I am a triangle," because the definition of "triangle" is "three-sided polygon". A truth table makes it clear that S and the converse of S are not logically equivalent, unless both terms imply each other:
A method of encoding mathematical and logical symbols and expressions as natural numbers, introduced by Kurt Gödel as part of his incompleteness theorems. Gödel sentence A self-referential sentence constructed in formal systems to demonstrate Gödel's incompleteness theorems, asserting its own unprovability within the system. Gödel-Dummett logic
In propositional logic, material implication [1] [2] is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not-or and that either form can replace the other in logical proofs.