Interval (mathematics)

ALGEBRA In Elementary Algebra , an interval is a set that contains every Real Number between two indicated numbers, and possibly the two numbers themselves. '''Interval notation''' is the notation in which permitted values for a Variable are expressed as ranging over a certain interval; " $5 < x < 9$ " is an example of the application of interval notation. In conventional interval notation, Parentheses ( $()$ ) indicate exclusion while Square Brackets ( $) indicate inclusion. For example, the interval " (10,20) " indicates the set of all real numbers between 10 and 20 but does ''not'' include 10 or 20, the first and last numbers of the interval, respectively. On the other hand, the interval " [10,20$ " includes both every number between $10$ and $20$ ''as well as'' $10$ and $20$ . Other possibilities are listed below. HIGHER MATHEMATICS In higher $S$ of a Totally Ordered Set $T$ with the property that whenever $x$ and $y$ are in $S$ and $x then z is in S .$ As mentioned above, a particularly important case is when $T=\mathbb{R}$ , the set of Real Number s. Intervals of $\mathbb{R}$ are of the following eleven different types (where $a$ and $b$ are real numbers, with $a < b$ ):
	# <math>	"a,b" class="copylinks" target="_blank">{Link without Title} =\{x\,\,a\leq x\leq b\}</math>
	# <math>[a,b)	\{x\,\,a\,\leq x<b\}</math>
	# <math>(a,b]	\{x\,\,a<x\leq b\}</math>
	# <math>(a,\infty)	\{x\,\,x>a\}</math>
	# <math>[a,\infty)	\{x\,\,x\geq a\}</math>
	# <math>(-\infty,b)	\{x\,\,x<b\}</math>
	# <math>(-\infty,b]	\{x\,\,x\leq b\}</math>
	:	"''a''," class="copylinks" target="_blank">''b'' = { ''x'' ''a'' &le ''x'' &le ''b'' }
	:T &middot S	{ ''x'' there is some ''y'' in ''T'', and some ''z'' in ''S'', such that ''x'' = ''y'' &middot ''z'' }
	<nowiki>]</nowiki>''a'',''b''<nowiki>[</nowiki>	{ ''x'' ''a'' < ''x'' < ''b'' }
		"''a'',''b''" class="copylinks" target="_blank">{Link without Title} = { ''x'' ''a'' &le ''x'' &le ''b'' }
	<nowiki>[</nowiki>''a'',''b''<nowiki>[</nowiki>	{ ''x'' ''a'' &le ''x'' < ''b'' }
	<nowiki>]</nowiki>''a'',''b''<nowiki>]</nowiki>	{ ''x'' ''a'' < ''x'' &le ''b'' }

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Interval (mathematics)