What does this square bracket and parenthesis bracket notation mean first1 last1

Question

I have seen number ranges represented as  first1 last1  and  first2 last2    I would like to know what such a notation means

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It can be a mathematical convention in the definition of an interval where square brackets mean  extremal inclusive  and round brackets  extremal exclusive

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That s a half-open interval    A closed interval  a b  includes the end points  An open interval  a b  excludes them    In your case the end-point at the start of the interval is included  but the end is excluded  So it means the interval  first1  lt   x  lt  last1     Half-open intervals are useful in programming because they correspond to the common idiom for looping   for  int i   0  i  lt  n    i             Here i is in the range  0  n

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The concept of interval notation comes up in both Mathematics and Computer Science   The Mathematical notation            denotes the domain  or  range  of an interval    The brackets   and   means    The number is included  This side of the interval is closed   The parenthesis   and   means     The number is excluded   This side of the interval is open     An interval with mixed states is called  half-open    For example  the range of consecutive integers from 1    10  inclusive  would be notated as such     1 10    Notice how the word inclusive was used  If we want to exclude the end point but  cover  the same range we need to move the end-point     1 11    For both left and right edges of the interval there are actually 4 permutations    1 10      2 3 4 5 6 7 8 9       Set has  8 elements  1 10      2 3 4 5 6 7 8 9 10    Set has  9 elements  1 10    1 2 3 4 5 6 7 8 9       Set has  9 elements  1 10    1 2 3 4 5 6 7 8 9 10    Set has 10 elements   How does this relate to Mathematics and Computer Science   Array indexes tend to use a different offset depending on which field are you in    Mathematics tends to be one-based  Certain programming languages tends to be zero-based  such as C  C    Javascript  Python  while other languages such as Mathematica  Fortran  Pascal are one-based      These differences can lead to subtle fence post errors  aka  off-by-one bugs when implementing Mathematical algorithms such as for-loops   Integers  If we have a set or array  say of the first few primes   2  3  5  7   11  13  17  19  23  29    Mathematicians would refer to the first element as the 1st absolute  element  i e  Using subscript notation to denote the index    a1   2 a2   3   a10   29   Some programming languages  in contradistinction  would refer to the first element as the zero th relative element    a 0    2 a 1    3   a 9    29   Since the array indexes are in the range  0 N-1  then for clarity purposes it would be  nice  to keep the same numerical value for the range 0    N instead of adding textual noise such as a -1 bias   For example  in C or JavaScript  to iterate over an array of N elements a programmer would write the common idiom of  i   0  i  lt  N with the interval  0 N  instead of the slightly more verbose  0 N-1     x000D   x000D  function main     x000D      var output       x000D      var a     2  3  5  7   11  13  17  19  23  29    x000D      for  var i   0  i  lt  10  i         0 10  x000D         output          i           a i      n   x000D   x000D      if  typeof window      undefined      Node command line x000D          console log  output   x000D      else x000D          document getElementById  output1   innerHTML   output  x000D    x000D    lt html gt  x000D        lt body onload  main     gt  x000D            lt pre id  output1  gt  lt  pre gt  x000D        lt  body gt  x000D    lt  html gt  x000D   x000D   x000D    Mathematicians  since they start counting at 1  would instead use the i   1  i  lt   N nomenclature but now we need to correct the array offset in a zero-based language   e g    x000D   x000D  function main     x000D      var output       x000D      var a     2  3  5  7   11  13  17  19  23  29    x000D      for  var i   1  i  lt   10  i         1 10  x000D         output          i           a i-1      n   x000D   x000D      if  typeof window      undefined      Node command line x000D          console log  output   x000D      else x000D          document getElementById   output2    innerHTML   output  x000D    x000D   lt html gt  x000D       lt body onload  main     gt  x000D           lt pre id  output2  gt  lt  pre gt  x000D       lt  body gt  x000D   lt  html gt  x000D   x000D   x000D    Aside   In programming languages that are 0-based you might need a kludge of a dummy zero th element to use a Mathematical 1-based algorithm  e g  Python Index Start  Floating-Point  Interval notation is also important for floating-point numbers to avoid subtle bugs   When dealing with floating-point numbers especially in Computer Graphics  color conversion  computational geometry  animation easing blending  etc   often times normalized numbers are used  That is  numbers between 0 0 and 1 0   It is important to know the edge cases if the endpoints are inclusive or exclusive     0 1    1e-M    0 999     0 1    1e-M    1 0  0 1    0 0    0 999     0 1    0 0    1 0   Where M is some machine epsilon   This is why you might sometimes see const float EPSILON   1e-  idiom in C code  such as 1e-6  for a 32-bit floating point number  This SO question Does EPSILON guarantee anything  has some preliminary details   For a more comprehensive answer see FLT EPSILON and David Goldberg s What Every Computer Scientist Should Know About Floating-Point Arithmetic  Some implementations of a random number generator  random   may produce values in the range 0 0    0 999    instead of the more convenient 0 0    1 0   Proper comments in the code will document this as  0 0 1 0  or  0 0 1 0  so there is no ambiguity as to the usage   Example    You want to generate random   colors   You convert three floating-point values to unsigned 8-bit values to generate a 24-bit pixel with red  green  and blue channels respectively  Depending on the interval output by random   you may end up with near-white  254 254 254  or white  255 255 255           -------- -----        random   Byte         -------- -----        0 999     254    lt -- error introduced       1 0       255         -------- -----    For more details about floating-point precision and robustness with intervals see Christer Ericson s Real-Time Collision Detection  Chapter 11 Numerical Robustness  Section 11 3 Robust Floating-Point Usage

User · Answer

A bracket -   or   - means that end of the range is inclusive -- it includes the element listed  A parenthesis -   or   - means that end is exclusive and doesn t contain the listed element  So for  first1  last1   the range starts with first1  and includes it   but ends just before last1  Assuming integers    0  5    1  2  3  4  0  5    1  2  3  4  5  0  5    0  1  2  3  4  0  5    0  1  2  3  4  5

[mathematical-notation] What does this square bracket and parenthesis bracket notation mean [first1,last1)?

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