In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a property is any characteristic that applies to a given
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
.
, a property ''p'' defined for all elements of a set ''X'' is usually defined as a function ''p'': ''X'' → , that is true whenever the property holds; or, equivalently, as the
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of ''X'' for which ''p'' holds; i.e. the set ; ''p'' is its
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
. However, it may be objected that the rigorous definition defines merely the
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (proof theory)
* Extension (predicate logic), the set of tuples of values that ...
of a property, and says nothing about what causes the property to hold for exactly those values.
Examples
Of
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
:
*
Parity is the property of an
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
of whether it is even or odd
For more examples, see
:Algebraic properties of elements.
Of
operations:
*
associative property
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
*
commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...
of
binary operations
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operation o ...
between
real and
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s
*
distributive property
In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary ...
For more examples, see
:Properties of binary operations.
See also
*
Unary relation
In mathematics, a finitary relation over a sequence of sets is a subset of the Cartesian product ; that is, it is a set of ''n''-tuples , each being a sequence of elements ''x'i'' in the corresponding ''X'i''. Typically, the relation descri ...
References
Mathematical terminology
Mathematical relations
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