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Q: Origin/etymology of Minimum, Maximum, Inclusive, Exclusive ( Answered,   0 Comments )
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 Subject: Origin/etymology of Minimum, Maximum, Inclusive, Exclusive Category: Science > Math Asked by: siam-ga List Price: \$5.00 Posted: 09 Apr 2005 01:56 PDT Expires: 09 May 2005 01:56 PDT Question ID: 507110
 ```As a programmer I've ran into instances of confusion when ascribing attributes of minExclusive and maxInclusive to data. I'm curious to know if there are intuitive or counter-intuitive relationships between all 4: minInclusive, minExclusive, maxExclusive, maxInclusive Does M*in*imum originally relate to *In*clusive, and does M*ax*imum originally relate to *Ex*clusive? Variation of this relationship may explain the confluence in combinations. Other resources on similar semantic stumbling blocks for programmers, or in general would be appreciated. I hope others may find this interesting as well.``` Request for Question Clarification by mathtalk-ga on 10 Apr 2005 10:12 PDT ```Hi, siam-ga: I don't believe there are direct etymological relationships between the dual notions of minimum/maximum and the dichotomous notions of inclusive/exclusive, though certainly a logical relationship can be proven, with perhaps a bit of stretching. However I'm interested to know what you mean, generally, by "ascribing attributes of minExclusive and maxInclusive to data." Are these respectively the smallest element _not_ in a data set and the biggest element in a data set? Such definitions make sense for a finite population (sampled by a data set) for which a linear ordering exists, but I'm not aware of any "standardized" vocabulary for these notions. regards, mathtalk-ga```
 ```Hi, siam-ga: The prefixes "in-" and "ex-" have come down to us in both English and French from Latin prepositions. Some discussion of "ex-" was already given in an earlier Google Answers thread: [Q: What does the "ex-" stand for? (Google Answers)] http://answers.google.com/answers/threadview?id=437977 where Google Answers Researcher gentryunderwood-ga noted that in Latin "ex" means "out(side) of": [ex- (one entry, Online Etymological Dictionary)] http://www.etymonline.com/index.php?term=ex- Especially in Latin the preposition has a secondary connotation of "from". For example, creation "ex nihilo", literally "out of nothing", intends to convey creation "from nothing". The English meaning "in(side)" or "(with)in" of prefix "in-" (sometimes "im-") is familiar as the preposition still means largely the same thing in English as it did in Latin, although in French the modified spelling "en-" (or "em-") is more common (and we have "embraced" many such words in English also). [in- (two entries, Online Etymological Dictionary)] http://www.etymonline.com/index.php?term=in- A confusing thing is that "in-" can also mean "not", in both Latin and English, derived from the Greek prefix "a(n)-", as well "un-". However to be brief, "include" and "exclude" are derived by prefixing from the Latin verb "claudere", meaning to close or shut, to meaning "keep in" or "keep out": [include] http://www.etymonline.com/index.php?term=include [exclude] http://www.etymonline.com/index.php?term=exclude * * * * * * * * * * * * * * * * * * * The distinction between inclusion and exclusion presents one with the seminal boundary betwee what is accepted and what is rejected. Such boundaries, both in absolute and relative contexts, form the environment for logic and reason. So it is reasonable to double check for hidden connections as we encounter them in programming and mathematics. * * * * * * * * * * * * * * * * * * * The words minimum and maximum (plurals minima and maximum) also have evident Latin origins. From the Latin root "min-" meaning small, we get "minus" for a deficit, the comparative "minor" for smaller, and the superlative "minimus" (neut. "minimum") meaning least: [minus] http://www.etymonline.com/index.php?term=minus [minimum] http://www.etymonline.com/index.php?term=minimum Similarly maximum is the Latin neutral superlative to a root "magnus" meaning great or large: [magnum] http://www.etymonline.com/index.php?term=magnum [maximum] http://www.etymonline.com/index.php?term=maximum * * * * * * * * * * * * * * * * * * * In an earlier Comment I pointed out that while inclusion and exclusion are dichotomous ("you're either in or out"), the relationship between minimum and maximum is not that of dichotomy, as in particular something could in theory be both at the same time. I referred instead to these as dual concepts, an approach admittedly influence by mathematical training, since one obtains one from the other by reversing the "sense" (direction) of the implied notion of inequality (comparison). At the level of mathematical logic one can often make out, as here, that the "dual" is a non-trivial form of double negative. Suppose for simplicity that our context is a linearly ordered finite set. The "included" maximum is that value in the set which is greater than any other element. It is at the same time the only element of the set which is _not_ less than some other element of the set. Likewise the "included" minimum can be defined as the value in the set which is less than any other element, or alternatively as the only element of the set which is _not_ bigger than some other element of the set. If we switch to a discussion or definition of "excluded" maximum or minimum, we find that there isn't much more that can be said to generally connect these with their "included" counterparts (except to say in a finite context as here that they are necessarily unequal, assuming a value cannot both be in and out of the set). In mathematical analysis there are contexts, inherently infinite such as the real numbers/Dedekind cuts on the rationals, in which it is important to prove equality between the least upper bound of a strictly increasing sequence {x_i} and the minimum (greatest lower bound) of the complement of the union of half open intervals (-oo, x_i). To fully explore these possibilities would lead us on a tangential discussion from which we'd not soon return. * * * * * * * * * * * * * * * * * * * We can push the derivations of words further back into the Indo-European or proto-Indo-European family of languages. The Latin/Greek roots "mag-"/"mega-" trace back to the Sanskrit "mahat" for great: [m:ht:a (mahataa) = great (Sanskrit)] http://www.alkhemy.com/sanskrit/dict/dictall.html or even these apparently related roots from the Dravidic family of languages: [235. magh = might, power (Dravidian)] http://www.datanumeric.com/dravidian/page060.html [247. me III, megh = great (Dravidian)] http://www.datanumeric.com/dravidian/page061.html The Latin/Greek roots "min-"/"mei-" may well trace back to the Dravidian "mei" for little, lessen: [250. mei I = little, lessen (Dravidian)] http://www.datanumeric.com/dravidian/page062.html It is easy to find echoes of our many modern prefixes for "not" in these ancient languages, niH in Sanskrit and ne in Dravidian: [en:H (niH) = without (Sanskrit)] http://www.alkhemy.com/sanskrit/dict/dictall.html [285. ne = negation (Dravidian)] http://www.datanumeric.com/dravidian/page065.html However I found no evidence that the dichotomy of in-/ex- prefixes or prepositions would "unify" in some way with the distinction between small and great at any stage of recorded language, short of degenerating to a common connection with not/negation/without. regards, mathtalk-ga```