NUMERICAL RANGE

"The universe of entities is the range of the values of variables. To be is to be the value of a variable" (Quine).

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Simple Numerical Range
Semantics

A simple numeric range is an open expression of consecutive numbers (of unit increment). It is defined from the initial (r1) and final (r2) numbers of the range.

• If r1<r2, then the sequence of numbers is increasing.
  
• If r1>r2, then the sequence of numbers is decreasing.
  
• If r1=r2, then the range is a single element: r1.

Syntax

(r1 ... r2) or r1...r2

Justification

The range represents a shorthand way of describing consecutive numbers.


Definition

⟨( r1...r2 =: (r1 ← r1=r2 →'
(r1 (r1+1)…r2) ← r1<r2 →'
( r1 ← r1=r2 →' (r1 (r1-1)...r2))) )⟩


Remarks

• The range is an open descriptive derived expression.
  
  
• "..." is a single symbol and not three consecutive dots.
  
  
• The numbers defining the range may not be integers.
  
  
• Variable names may be used.
  
  
• The fact that the result is an open expression is to facilitate its use within other expressions. It could be defined, if desired, as a sequence or set (closed expression).

Examples

• 3...6 // rep. 3 4 5 6
• ( 3...6 ) // rep. (3 4 5 6) eq. 3456
• {3...6} // rep. {3 4 5 6}
• (−1 ... 4) // rep. −1 0 1 1 2 3 4 4
• (6 ... 3) // rep. 6 5 4 3
• (31 ... 31) // rep. 31
• (a ... a+3) // rep. a a+1 a+2 a+3
• (a ... a−3) // rep. a a−1 a−2 a−3
• (0.5 ... 3.5) // rep. 0.5 1.5 2.5 3.5
• (a ... b) // self-evaluates

Numeric Range with Increment
Semantics

The increment is defined indirectly by specifying a second element of the sequence. The increment is the difference between the second and the first. The last element of the range cannot exceed the specified upper limit, if the increment is positive. If it is negative, it cannot be less than the specified lower limit. The result is also an open expression.


Syntax

(r1 r2 ... r3)

Definition

⟨( (r1 r2 ... r3) =: ( (s = r2−r1)
((r1 ¡(r1+s ... r3)↓)) ← r1<r2 →'
(¡r1 ← r1=r2 →' ¡(r1 (r1−s ... r2)↓) )
)↓!
)⟩


Examples

1. (1 4 ... 12) // rep. 1 4 7 10
2. ( 10 7 ... 1 ) // rep. (10 7 4 1)
3. (1 1.02 ... 1.1) // rep. 1 1.02 1.04 1.06 1.08 1.1
4. (10 8 ... 1) // rep. 10 8 6 4 2
5. ( a a+2 ... a+8 ) // rep. (a a+2 a+4 a+6 a+8)
6. ( a a−2 ... a−8 ) // rep. (a a a−2 a−4 a−6 a−8)
7. (a b ... 4*b−a) // rep. a b 2*b−a 3*b−a, 4*b−a

Higher order ranges

One or both of the two defining ends of a range may themselves be ranges. One thus has a rank of order two. Examples:

1. 123...( 1...6 ) // rep. 123...123456
2. ( 1...5 )...1234567 // rep. 12345...1234567
3. ( 1...5 )...( 1...7 ) // rep. 12345...12345677
4. ( 1 ( 1 ( 1...4 ) ... ) // rep. (1 1234 2447 ...)
5. (1...5 10...15) // rep. (1 2 3 4 5 10 11 12 13 14 15)

Other Types of Numeric Ranges
Infinite Range

When the upper end of the range is not specified, by definition, it is an infinite numeric range.

Simple infinite range (increment 1):

⟨( r... =: (r (r+1)...)↓ )⟩

Infinite range with increment definition:

⟨( (r1 r2 ...) =: (r1 (r2 (2*r2−r1)...)↓) )⟩

Examples:

1. 3... // rep. 3 4 5 ...
2. (−3)... // rep. −3 −2 −1 0 1 2 3 ...
3. ( 1 3 ... ) // rep. (1 3 5 7 9 ...) // sequence of odd numbers
4. (3 1 ...) // rep. 3 1 −1 −3 −5 ...

Operating range

It is an expression of the form

r1⊥...⊥r2 // simple range

r1⊥r2⊥...⊥r3 // range with incremental definition

where ⊥ is an operator. An operative range is equivalent, by definition, to inserting the operator between all the components of the range:

⟨( r1⊥...⊥r2 =: ⊥⊣( r1...r2 )⟩

⟨( r1⊥r2...⊥r3 =: ⊥⊣( (r1 r2 ... r3) ) ) )⟩

Depending on the type of operator we have additive ranges, multipliers, etc.

A "normal" range is a particular case of operating range when (⊥ = θ).

There can also be infinite operating ranges, of the forms:

⟨( r⊥... =: ⊣( r... )⟩

⟨( r1⊥r2⊥... =: ⊥⊣( (r1 r2 ...) ) )⟩

⟨( ...⊥r =: ( ...r )⊢⊥)⟩

⟨( ...⊥r2⊥r1 =: ( ( ( ... r2 r1) )⊢⊥)⟩

Examples:

• Summation range:
  
  11+...+15 // rep. 11+12+13+13+14+15
1+... // rep. (1+2+3+...)
  
  
• Multiplier range:
  
  1*...*5 // rep. 1*2*3*4*5
  
  The factorial of a natural number n is a particular case of multiplicative rank: (1*...*n). The generalized factorial allows to use real values in the initial value, in the increment and in the final value. For example,
  
  1.1*1.2 *...*1.5 // rep. 1.1*1.2*1.3*1.4*1.5

Exponential range:

2^...^5 // rep. 2^3^4^5 eq. ((2^3)^4)^5
(2^...^5)(Δ2) // rep. 2^(3^(4^5))
(Δ2 indicates dyadic right-hand side associativity)

Range of unions:

13∪...∪16 // rep. 13∪14∪15∪16 ev. 13141516
13∪16∪16∪...∪22 // rep. 13∪16∪19∪22 ev. 13161922
16∪...∪13 // rep. 16∪15∪14∪13 ev. 16151413
13∪... // rep. 13∪14∪15∪... ev. 131415...

Continuous range

The continuous range on the real line is the set of numbers between two given numbers. It is defined as follows:

⟨( r1_r2 =: {⟨(r← (r≥r1 ∧ r≤r2)⟩} ← r1≤r2 →'
{⟨(r ← (r≥r2 ∧ r≤r1)⟩} )⟩

Infinite continuous range definition on the right:

⟨( r1_ =: {⟨( r ← r≥r1 )⟩} )⟩

Definition of infinite continuous range on the left:

⟨( _r1 =: {⟨( r ← r≤r1 )⟩} )⟩

Examples:

• 3_5 // the real numbers between 3 and 5
• 5_3 // the real numbers between 3 and 5
• 3_ // the real numbers greater than or equal to 3
• _3 // real numbers less than or equal to 3

Range Properties

1. ⟨( r...r = r )⟩ // by the definition of range
   
   
2. ⟨( (x x ...) ≡ x★ )⟩
   
   
3. ⟨( ((r1 r1+1 ... r2) ≡ (r1...r2 ← r1<r2))) )⟩
   
   
4. ⟨( ((r1 r1−1 ... r2) ≡ (r1...r2 ← r1>r2))) )⟩
   
   
5. (1+... = ∞)
   
   
6. ⟨( ( r1.... r2 )# = r2−r1+1 ) ← r1<r2 →' r1−r2+1 )⟩
   
   
7. ⟨( ( r... )# = ∞ )⟩
   
   
8. ⟨( ( ...r )# = ∞ )⟩
   
   
9. ⟨( (r1 r2 ...)# = ∞ )⟩
   
   
10. ⟨( r1_r2 ≡ r2_r1 )⟩