Difference between revisions of "Half-open interval"
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A '''half-open interval''' is an [[interval]] which has either a [[maximum]] or a [[minimum]] element but not both. | A '''half-open interval''' is an [[interval]] which has either a [[maximum]] or a [[minimum]] element but not both. | ||
| − | If a half-open interval has a minimum <math>a</math> but no maximum, then it is denoted by <math>[a,b)</math>, where <math>b</math> is the [[Least upper bound|supremum]], or <math>\infty</math> if no supremum exists. Alternatively, <math>[a,b)</math> is the [[set]] of all <math>x</math> such that <math>a \leq x</math> and <math>x < b</math>. | + | If a half-open interval has a minimum <math>a</math> but no maximum, then it is denoted by <math>[a,b)</math>, where <math>b</math> is the [[Least upper bound|supremum]] (least upper bound), or <math>\infty</math> if no supremum exists. Alternatively, <math>[a,b)</math> is the [[set]] of all <math>x</math> such that <math>a \leq x</math> and <math>x < b</math>. |
| − | If a half-open interval has a maximum <math>b</math> but no minimum, then it is denoted by <math>(a,b]</math>, where <math>a</math> is the [[Greatest lower bound|infimum]], or <math>-\infty</math> if no infimum exists. Alternatively, <math>(a,b]</math> is the set of all <math>x</math> such that <math>a < x</math> and <math>x \leq b</math>. | + | If a half-open interval has a maximum <math>b</math> but no minimum, then it is denoted by <math>(a,b]</math>, where <math>a</math> is the [[Greatest lower bound|infimum]] (greatest lower bound), or <math>-\infty</math> if no infimum exists. Alternatively, <math>(a,b]</math> is the set of all <math>x</math> such that <math>a < x</math> and <math>x \leq b</math>. |
==Examples== | ==Examples== | ||
Latest revision as of 15:12, 5 March 2022
A half-open interval is an interval which has either a maximum or a minimum element but not both.
If a half-open interval has a minimum 
but no maximum, then it is denoted by 
, where 
is the supremum (least upper bound), or 
if no supremum exists. Alternatively, is the set of all 
such that 
and 
.
If a half-open interval has a maximum but no minimum, then it is denoted by ![$(a,b]$](http://latex.artofproblemsolving.com/c/d/f/cdf3085063293f29962808ba253dec1faa9866fb.png)
, where is the infimum (greatest lower bound), or 
if no infimum exists. Alternatively, is the set of all such that 
and 
.
Examples
is a half-open interval with a minimum but no maximum.
![$(-1,1]$](http://latex.artofproblemsolving.com/e/1/4/e147ba06933a19883fc3a01f29328f188b9f6fdb.png)
is a half-open interval with a maximum but no minimum.
, the set of nonnegative real numbers, is a half-open interval with no supremum.
![$(-\infty,0]$](http://latex.artofproblemsolving.com/5/0/d/50d25d3e486146edd225ea505aaa53016f41edb7.png)
, the set of nonpositive real numbers, is a half-open interval with no infimum.
See also
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