Intervals

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A (real) interval is a set of real numbers with the property that any number that lies between the lower bound (when left-bounded) and the upper bound (when right-bounded) of the set is also included in the set, thus it is connected (which implies convex, since it is one-dimensional).

Lower and upper bounds

The lower bound is called either

• the minimum (if included),
• the infimum (if excluded),

while the upper bound is called either

• the maximum (if included);
• the supremum (if excluded).

An interval is left-closed, i.e. contains all its limit points to the left, if it is either

• a left-unbounded interval or the unbounded interval (since it contains all its limit points to the left);
• a left-closed [right-unbounded or bounded] interval.

An interval is right-closed, i.e. contains all its limit points to the right, if it is either

• a right-unbounded interval or the unbounded interval (since it contains all its limit points to the right);
• a right-closed [left-unbounded or bounded] interval.

A closed interval is left-closed and right-closed, i.e. contains all its limit points.

An open interval is left-open and right-open.

Degenerate intervals

A degenerate interval has its upper bound equal to its lower bound and is either

• a singleton $[a,a]=\{a\}$;
• the empty set $(a,a)=\emptyset$.

Bounded intervals

Notation (standard notation in the left column, nonstandard notation in the middle column) for bounded intervals:

${\begin{aligned}(a,b)={\mathopen {]}}a,b{\mathclose {[}}&=\{x\in \mathbb {R} \,|\,a<x<b\},\\{}[a,b)={\mathopen {[}}a,b{\mathclose {[}}&=\{x\in \mathbb {R} \,|\,a\leq x<b\},\\{}(a,b]={\mathopen {]}}a,b{\mathclose {]}}&=\{x\in \mathbb {R} \,|\,a<x\leq b\},\\{}[a,b]={\mathopen {[}}a,b{\mathclose {]}}&=\{x\in \mathbb {R} \,|\,a\leq x\leq b\}.\end{aligned}}$

A bounded interval is

• an open interval, denoted $(a,b)$, which is the set of all real numbers between $a$ and $b$, exclusively;
• an open-closed half-open interval (resp., half-closed interval), denoted $(a,b]$, which excludes $a$;
• a closed-open half-open interval (resp., half-closed interval), denoted $[a,b)$, which excludes $b$; or
• a closed interval, denoted $[a,b]$, which is the set of all real numbers from $a$ to $b$, inclusively.

All [nondegenerate] bounded intervals have the same cardinality as the unit interval, since we have the linear maps

$x\in (0,1)\mapsto a+x\,(b-a)\in (a,b),$

$y\in (a,b)\mapsto {\frac {y-a}{b-a}}\in (0,1),$

which are bijective, i.e. one-to-one and onto.

Half-bounded intervals

Notation (standard notation in the left column, nonstandard notation in the middle column) for half-bounded intervals:

${\begin{aligned}(a,+\infty )={\mathopen {(}}a,+\infty {\mathclose {[}}&=\{x\in \mathbb {R} \,|\,a<x\},\\{}[a,+\infty )={\mathopen {[}}a,+\infty {\mathclose {[}}&=\{x\in \mathbb {R} \,|\,a\leq x\},\\{}(-\infty ,b)={\mathopen {]}}\!-\infty ,b{\mathclose {)}}&=\{x\in \mathbb {R} \,|\,x<b\},\\{}(-\infty ,b]={\mathopen {]}}\!-\infty ,b{\mathclose {]}}&=\{x\in \mathbb {R} \,|\,x\leq b\},\\{}\end{aligned}}$

A half-bounded interval (half-unbounded interval), also called ray or half-line, is either

• a left-bounded interval (right-unbounded interval), which is either
  • a left-open interval (left-open ray or left-open half-line): $(a,+\infty )=\{x\,|\,x>a\}$,
  • a left-closed interval (left-closed ray or left-closed half-line): $[a,+\infty )=\{x\,|\,x\geq a\}$ (a closed interval, since it contains all its limit points);
• a left-unbounded interval (right-bounded interval), which is either
  • a right-open interval (right-open ray or right-open half-line): $(-\infty ,b)=\{x\,|\,x<b\}$,
  • a right-closed interval (right-closed ray or right-closed half-line): $(-\infty ,b]=\{x\,|\,x\leq b\}$ (a closed interval, since it contains all its limit points).

Unbounded interval

The unbounded interval is the set of real numbers itself (a closed interval, since it contains all its limit points)

$(-\infty ,+\infty )={\mathopen {]}}\!-\infty ,+\infty {\mathclose {[}}=\mathbb {R} .$

The real line (continuum) has the same cardinality as the unit interval, since

$x\in (0,1)\mapsto \log \left({\frac {x}{1-x}}\right)\in \mathbb {R} ,$

$y\in \mathbb {R} \mapsto {\frac {1}{1+e^{-y}}}\in (0,1),$

which are bijective, i.e. one-to-one and onto, strictly increasing maps.

See also

• Unit interval

External links

• Weisstein, Eric W., Interval, from MathWorld—A Wolfram Web Resource.