In algebraic-operation sequences {mathematical function} {function, mathematics}, every domain value can result in only one range value. Defined variable {dependent variable, function} equals function of original variable {independent variable, function}: y = f(x). Functions can have more than one independent variable: y = f(x1, x2, ...).
domain
Independent variable value is in a possible-value set.
range
Dependent variable value is in a possible-value set.
mapping
Domain elements correspond to one range element {image, range} {mapping, range}: x -> y.
one-to-one correspondence
Domain element can correspond to only one range element, and range elements can correspond to only one domain element.
relation
In algebraic-operation sequences {relation, function}, every domain value can result in one or more range values. Independent-variable values can result in more than one dependent-variable value. Relations are about order, symmetry, transitivity, reflexivity, equivalence, material implication, inclusion, one-to-one correspondence, and many-to-one correspondence relations.
relation: explicit
Two variables can directly relate {explicit relation} {explicit function}. Example is y = 3*x.
relation: implicit
Two variables can relate to same parameter {implicit relation} {implicit function, algebra}. Example is x = t and y = 3*t, so y = 3*x.
relation: reflexive
Symbol, concepts, or sentences can refer to itself. Example is "This sentence is true." Relations can relate domain elements to themselves {reflexive relation}. Example is the equality relation x = x.
relation: anti-reflexive
Relations can relate no domain element to itself {anti-reflexive relation}. Example is the inequality relation: y > x.
relation: non-reflexive
One or more domain elements can not relate to itself {non-reflexive relation}.
relation: recursive
Relations can repeatedly apply algebraic-operation sequences to domain elements to make successive terms {recursive relation} {recurring sequence} {recurring series}: for example, x, x + x = 2*x, 2*x + x = 3*x, ..., (n - 1)*x + x = n*x. In recursive relations, nth term has coefficient {scale, relation}.
relation: transitive
Relations {transitive relation} can preserve value order. Example is if a > b and b > c, then a > c.
variation: monotonic decreasing
Quantities can always decrease or stay the same {monotonic decreasing}.
variation: monotonic increasing
Quantities can always increase or stay the same {monotonic increasing}.
variation: joint
Variables can vary directly with product of other variables {joint variation}.
variation: combined
Variables can relate to expressions of other variables {combined variation}.
variation: invariance
After transformation, functions can result in same product as before. By energy conservation, functions that calculate energy have invariance. Invariants are covariants of order zero.
Mathematical Sciences>Algebra>Function
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Date Modified: 2022.0224