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A function that takes a single argument as input, such as () =, is called a unary function. A function of two or more variables is considered to have a domain consisting of ordered pairs or tuples of argument values. The argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolic angle.
In proof by contradiction, also known by the Latin phrase reductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, a logical contradiction occurs, hence the statement must be false. A famous example involves the proof that is an irrational number:
A typical example is the proof of the proposition "there is no smallest positive rational number": assume there is a smallest positive rational number q and derive a contradiction by observing that q / 2 is even smaller than q and still positive.
Reductio ad absurdum, painting by John Pettie exhibited at the Royal Academy in 1884. In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum (Latin for "argument to absurdity") or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.
The logical form of this argument is known as modus ponens, [39] which is a classically valid form. [40] So, in classical logic, the argument is valid, although it may or may not be sound, depending on the meteorological facts in a given context. This example argument will be reused when explaining § Formalization.
ad – adjoint representation (or adjoint action) of a Lie group. adj – adjugate of a matrix. a.e. – almost everywhere. AFSOC - Assume for the sake of contradiction; Ai – Airy function. AL – Action limit. Alt – alternating group (Alt(n) is also written as A n.) A.M. – arithmetic mean. AP – arithmetic progression. arccos ...
In the context of limits, this is shorthand meaning for sufficiently large arguments; the relevant argument(s) are implicit in the context. As an example, the function log(log(x)) eventually becomes larger than 100"; in this context, "eventually" means "for sufficiently large x." factor through
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. This is a listing of articles which explain some of these functions in more detail.