Math 2.0

definition

Interval

Definition

A subset II of R\mathbb{R} is an interval when, for all xx and yy in II and each real zz with x≤z≤yx \le z \le y, also z∈Iz \in I. For real numbers a≤ba \le b, the notation is: [a,b]={x∈R:a≤x≤b},(a,b)={x∈R:a<x<b},[a, b] = \{x \in \mathbb{R} : a \le x \le b\}, \qquad (a, b) = \{x \in \mathbb{R} : a < x < b\}, [a,b)={x∈R:a≤x<b},(a,b]={x∈R:a<x≤b}.[a, b) = \{x \in \mathbb{R} : a \le x < b\}, \qquad (a, b] = \{x \in \mathbb{R} : a < x \le b\}. The set [a,b][a, b] is a closed interval and (a,b)(a, b) is an open interval. For each real aa, the notation is [a,∞)={x:x≥a}[a, \infty) = \{x : x \ge a\}, (a,∞)={x:x>a}(a, \infty) = \{x : x > a\}, (−∞,a]={x:x≤a}(-\infty, a] = \{x : x \le a\} and (−∞,a)={x:x<a}(-\infty, a) = \{x : x < a\}.

Remarks

By the completeness axiom, each interval is one of the sets above, or R\mathbb{R}, or ∅\emptyset. In some countries, the notation ]a,b[]a, b[ replaces (a,b)(a, b). The symbols ∞\infty and −∞-\infty are not real numbers. They show only that the interval has no bound on that side.

Explanations

L1 human

An interval is a piece of the number line without holes. A square bracket means that the end point belongs to the interval, and a round bracket means that it does not. For example, [0,1)[0, 1) contains 00 and 0.9990.999, but not 11.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 9 nodes in this order. The foundations come first, and the list ends with this node.

Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionRational numbers, object, no evidence levelRational numbersobjectRing of integers, statement, no evidence levelRing of integersstatementThe rationals form a field, statement, no evidence levelThe rationals form a …statementThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementOrder on the reals, definition, no evidence levelOrder on the realsdefinitionInterval, definition, no evidence levelIntervaldefinition
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Integers object
  3. Ordered field definition
  4. Rational numbers object
  5. Ring of integers statement
  6. The rationals form a field statement
  7. The rationals form an ordered field statement
  8. Order on the reals definition
  9. Interval definition

What depends on this

172 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Interval_(mathematics), Definition of an interval
  2. Web https://en.wikipedia.org/wiki/Interval_(mathematics), Notations for intervals

What links here

Used by