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In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.
Let (,) and (,) be ordered pairs. Then the characteristic (or defining) property of the ordered pair is: (,) = (,) = =.. The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called the Cartesian product of A and B, and written A × B.
Now, we can impose relations among symbols in X by taking a quotient. Explicitly, if E is a subset of F, then the quotient ring of F by the ideal generated by E is called the ring with generators X and relations E. If we used a ring, say, A as a base ring instead of , then the resulting ring will be over A.
This broad definition of a function encompasses more relations than are ordinarily considered functions in contemporary mathematics. For example, Hardy's definition includes multivalued functions and what in computability theory are called partial functions .
All definitions tacitly require the homogeneous relation be transitive: for all ,,, if and then . A term's definition may require additional properties that are not listed in this table. Fig. 1 The Hasse diagram of the set of all subsets of a three-element set { x , y , z } , {\displaystyle \{x,y,z\},} ordered by inclusion .
A predicate is a statement or mathematical assertion that contains variables, sometimes referred to as predicate variables, and may be true or false depending on those variables’ value or values. In propositional logic, atomic formulas are sometimes regarded as zero-place predicates. [1] In a sense, these are nullary (i.e. 0-arity) predicates.
In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity , which describes a relation that is never transitive.
Comparability with respect to induces a canonical binary relation on ; specifically, the comparability relation induced by is defined to be the set of all pairs (,) such that is comparable to ; that is, such that at least one of and is true.