# BPFK Section: Intervals

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### cmavo bi'i (BIhI)

#### Proposed Definition

Non-logical interval connective. Joins two values into an unordered interval between the two values.

• {bi'o}

• between

#### Usage Examples

mi sanli la .drestyn. bi'i la .frankfurt.
I stand between Dresden and Frankfurt.

#### Notes

The order of the connected values does not matter.

### cmavo bi'o (BIhI)

#### Proposed Definition

Non-logical interval connective. Joins two values into an ordered interval that reaches from the first value to the second.

• {bi'i}

• from ... to

#### Usage Examples

mi gunka ze'a lo pavdei bi'o lo mumdei
I work for the time spanning from Monday to Friday.
mi gunka ze'a lo mumdei se bi'o lo pavdei
I work for the time spanning to Friday from Monday.

#### Notes

Order of the connected values very much so does matter. For (a, b) to be good ("positive"), it must be the case that ${\displaystyle a\leq b}$. Conventionally, one might define -(b, a) = (a, b) for all real numbers a, b; in this case (b, a) is considered to be bad ("negative") and requires the corrective factor of -1 in the evaluation of the expression.

### cmavo mi'i (BIhI)

#### Proposed Definition

Non-logical interval connective. Joins two values into an interval and refers to its center-range, or the range surrounding the center. The first value specifies the center, the second value specifies the distance from the center.

#### Proposed Keywords

• center-range
• surrounding center

#### Usage Examples

lo bamjakne ba janli la .moskov. mi'i lo ki'otre be li panono
The missile will hit within 100 kilometers of Moscow.
lo .teilr. zei sujypoi pe lo dugri fancu cu jbize'a ve'a li pa ga'omi'i pa
The Taylor series of the logarithm function converges within 1 of 1, closed at the center.

### cmavo ga'o (GAhO)

#### Proposed Definition

Inclusive interval. Indicates that a boundary is included in an interval. When used before bi'o or bi'i, it indicates that the beginning of the interval is closed. When used after bi'o or bi'i, it indicates that the end of the interval is closed. When used before mi'i, it indicates that the center of the interval is closed. When used after mi'i, it indicates that the outside of the interval is closed.

• {ke'i}

#### Proposed Keywords

• inclusive interval
• closed interval
• include boundaries

#### Usage Examples

mi sanli la .drestyn. ga'o bi'i ga'o la .frankfurt.
I am standing on the interval between Dresden and Frankfurt, inclusive of both.
mi sanli la .drestyn. ga'o bi'i ke'i la .frankfurt.
I am standing on the interval between Dresden (inclusive) and Frankfurt (exclusive).
mi sanli la .drestyn. ke'i bi'i ga'o la .frankfurt.
I am standing on the interval between Dresden (exclusive) and Frankfurt (inclusive).

### cmavo ke'i (GAhO)

#### Proposed Definition

Exclusive interval. Indicates that a boundary is excluded from an interval. When used before bi'o or bi'i, it indicates that the beginning of the interval is open. When used after bi'o or bi'i, it indicates that the end of the interval is open. When used before mi'i, it indicates that the center of the interval is open. When used after mi'i, it indicates that the outside of the interval is open.

• {ga'o}

#### Proposed Keywords

• exclusive interval
• open interval
• exlude boundaries

#### Usage Examples

mi sanli la .drestyn. ga'o bi'i ke'i la .frankfurt.
I am standing on the interval between Dresden (inclusive) and Frankfurt (exclusive).
mi sanli la .drestyn. ke'i bi'i ga'o la .frankfurt.
I am standing on the interval between Dresden (exclusive) and Frankfurt (inclusive).
mi sanli la .drestyn. ke'i bi'i ke'i la .frankfurt.
I am standing on the interval between Dresden and Frankfurt, exclusive of both.

### Proposed grammar changes

In the PEG grammar, replace the line
^ joik <- SE-clause? JOI-clause NAI-clause? / interval / GAhO-clause interval GAhO-clause
^ with the line
^ joik <- SE-clause? JOI-clause NAI-clause? / interval / GAhO-clause interval / interval GAhO-clause / GAhO-clause interval GAhO-clause ^ A joik cannot be immediately followed by another joik, so a sequence "BIhI GAhO BIhI" cannot occur, so determining to which BIhI a GAhO belongs does not depend on the order in this line.

### General Section Notes

BIhI may be used with only one GAhO next to it; this indicates that one end of the interval is closed or open, but says nothing about the other end. The Taylor series mentioned above diverges at 0, but converges at all other points on the boundary of the disk of convergence; thus the outside is neither closed nor open.

The definition of GAhO with mi'i differs from that in CLL, but there appears to be no prior usage of GAhO with mi'i.

Allowing BIhI with only one GAhO is a change in the grammar. Since GAhO is used only with BIhI, and no [GAhO] BIhI [GAhO] may be followed immediately by another [GAhO] BIhI [GAhO], this introduces no new ambiguity.

#### Interval or Point?

There is a question whether abu bi'i by denotes the interval [a..b] or some point in the interval. Here are some examples, showing which way it has been used:

lo kanla be lo remna cu viska loi gusni poi boxna folo navytre beli 400 bi'i li 800
"The human eye can see light of wavelength between 400 and 800 nanometers."

Wikipedia gusni
unclear

le trucerda be fi ty. ba'orgau le se sornaitru la madjarorsag. bi'i la jugygug.
"His successors expanded the empire from Hungary to China."

Wikipedia tcingis xan
interval

le critu dorduncte cu cabna le 22 bi'i 24 moi be le xlima'i
"The autumnal equinox occurs between the 22nd and the 24th of September."

Wikipedia dorduncte
point

pamoi mitre fo piki'oki'opa lo jedji'o bi'i lo mijycukla
"The first standard for the meter was one ten-millionth of the distance between the pole and the equator."

Wikipedia treci'e
interval

la sangug. stici la spetsias. bi'i la riminis.
"Spain is west of the La Spezia-Rimini line."

Wikipedia sanban
interval

pa jijyjutsi po'u la'o ly. Toona .ly. kuspe lo norgla jugygu'e bi'i lo nanstuna sralygu'e
"One genus, Toona, ranges from temperate China to southeast Australia."

Wikipedia spatrmeliace
interval

la'aru'e lo skalduna cu stali la .auskalerik ca'o le nu lo remna be la kromanion. cu basti lo remna be la neANdertal. kei bi'o le cabna
"Possibly, Basques have been in Basque Country from the time Cro-Magnon Man replaced Neanderthal Man until now."

Wikipedia skalduna
interval

srana lo tolci'o glibau noi pu renvi de'i li ji'i 0450 bi'o ji'i 1066
ko'e cu se banta'a co cabna lo tolci'o berdo'obau noi jibni ko'e lo ka ckini zi'e noi bangu lai vikingr. poi ke'a lo gligu'e de'i li 0790 bi'o ji'i 1040 gunta ce'o co'arxabju
"It was spoken at the same time as Old Norse, which is closely related and was spoken by the Vikings, who from 790 to 1040 attacked and then occupied England."
srana lo midju glibau noi pu renvi de'i li ji'i 1066 bi'o ji'i 1500
lo gligu'e cu jamna lo fasygu'e soi vo'a vo'e de'i li 1202 bi'o 1214
"England warred France and vice versa from 1202 to 1214."
srana lo purci ke cabna glibau noi pu renvi de'i li ji'i 1500 bi'o ji'i 1650
lo glico vlasrocku pe la oksford ge'u noi ke mulno je tcilykai ke'e pe'a pixra be ro glico valsi be ze'u lo ri citri cu se finti de'i li 1858 bi'o 1928
"The Oxford English Dictionary, a complete and detailed picture of all English words throughout their history, was authored from 1858 to 1928."

Wikipedia citri lo glibau
all intervals

by fy nitcu lo remna poi milxe bi'o banli to pa mai toi certu la lojban
"BF needs people who are slightly to greatly (firstly) expert in Lojban."

camgusmis
point

_____

lai .krtisfranks. maintains that these words can only refer to sets (intervals) and using them to select individual points from those implied sets (even in the case of [a, a] = {a}) amounts to abuse of 'notation'.