Difference between revisions of "Math notes / Geometry and its relatives"
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===Intervals===  ===Intervals===  
−  +  Intervals are ranges within the real number set.  
−  Intervals are ranges within the real number set.  +  
−  +  
+  They are set definitions which include all values between the mentioned numbers. Whether the specified numbers themselves are included depends on whether they are mentioned/denoted to be open or not:  
+  * '''open intervals''' exclude both, e.g. ''1 < x < 2''  
* '''closed intevals''' include both, e.g. ''1 ≤ x ≤ 2''  * '''closed intevals''' include both, e.g. ''1 ≤ x ≤ 2''  
* '''halfclosed'''/'''halfopen''' intervals include one and not the other, e.g. ''1 < x ≤ 2'' or ''1 ≤ x < 2''  * '''halfclosed'''/'''halfopen''' intervals include one and not the other, e.g. ''1 < x ≤ 2'' or ''1 ≤ x < 2''  
+  
* ''' ''both'' open and closed intervals''', as in the special cases of the empty/null set (∅), ''∞ < x < ∞'',  * ''' ''both'' open and closed intervals''', as in the special cases of the empty/null set (∅), ''∞ < x < ∞'',  
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* ''1<x≤2'' would be denoted as (1,2] and ]1,2], respectively  * ''1<x≤2'' would be denoted as (1,2] and ]1,2], respectively  
+  <!  
Notes:  Notes: 
Revision as of 13:56, 6 July 2020
This is more for overview of my own than for teaching or exercise.
Other data analysis, data summarization, learning

Contents
Intervals
Intervals are ranges within the real number set.
They are set definitions which include all values between the mentioned numbers. Whether the specified numbers themselves are included depends on whether they are mentioned/denoted to be open or not:
 open intervals exclude both, e.g. 1 < x < 2
 closed intevals include both, e.g. 1 ≤ x ≤ 2
 halfclosed/halfopen intervals include one and not the other, e.g. 1 < x ≤ 2 or 1 ≤ x < 2
 both open and closed intervals, as in the special cases of the empty/null set (∅), ∞ < x < ∞,
There are two different (and sometimes confusable) shorthand notations, one using round and square brackets (set builder notation), the other using only square brackets (ISO notation). Just stick to the one you are used to, but it can be handy to know about both.
For example:
 1<x<2 would be denoted as (1,2) and ]1,2[, respectively
 1≤x≤2 would be denoted as [1,2] and [1,2], respectively
 1<x≤2 would be denoted as (1,2] and ]1,2], respectively