Unit interval
In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital letter I). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology.
In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: (0,1], [0,1), and (0,1). However, the notation I is most commonly reserved for the closed interval [0,1].
Properties
The unit interval is a
is obtained by taking a topological product of countably many copies of the unit interval.In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1. Its standard orientation goes from 0 to 1.
The unit interval is a
Cardinality
The size or cardinality of a set is the number of elements it contains.
The unit interval is a subset of the real numbers . However, it has the same size as the whole set: the
The number of elements (either real numbers or points) in all the above-mentioned sets is uncountable, as it is strictly greater than the number of natural numbers.
Orientation
The unit interval is a curve. The open interval (0,1) is a subset of the positive real numbers and inherits an orientation from them. The orientation is reversed when the interval is entered from 1, such as in the integral used to define natural logarithm for x in the interval, thus yielding negative values for logarithm of such x. In fact, this integral is evaluated as a signed area yielding negative area over the unit interval due to reversed orientation there.
Generalizations
The interval [-1,1], with length two, demarcated by the positive and negative units, occurs frequently, such as in the
Sometimes, the term "unit interval" is used to refer to objects that play a role in various branches of mathematics analogous to the role that [0,1] plays in homotopy theory. For example, in the theory of quivers, the (analogue of the) unit interval is the graph whose vertex set is and which contains a single edge e whose source is 0 and whose target is 1. One can then define a notion of
Fuzzy logic
In logic, the unit interval [0,1] can be interpreted as a generalization of the Boolean domain {0,1}, in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 − x; conjunction (AND) is replaced with multiplication (xy); and disjunction (OR) is defined, per De Morgan's laws, as 1 − (1 − x)(1 − y).
Interpreting these values as logical
See also
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References
- Robert G. Bartle, 1964, The Elements of Real Analysis, John Wiley & Sons.