Etymology
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-let 
diminutive noun-forming element, Middle English, from Old French -elet, which often is a double-diminutive. It consists of Old French diminutive -et, -ette (see -et) added to nouns in -el, which in many cases represents Latin diminutive -ellus; see -el (2)). "The formation did not become common until the 18th c." [OED].
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let (n.)

"stoppage, obstruction" (obsolete unless in legal contracts), late 12c., from archaic verb letten "to hinder," from Old English lettan "hinder, delay, impede," etymologically "make late," from Proto-Germanic *latjan (source also of Old Saxon lettian "to hinder," Old Norse letja "to hold back," Old High German lezzen "to stop, check," Gothic latjan "to hinder, make late"), related to *lata-, source of late (adj.), from PIE root *‌‌lē- "to let go, slacken."

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let (v.)

Old English lætan (Northumbrian leta) "to allow; to leave behind, depart from; leave undone; bequeath," also "to rent, put to rent or hire" (class VII strong verb; past tense let, leort, past participle gelæten), from Proto-Germanic *letan (source also of Old Saxon latan, Old Frisian leta, Dutch laten, Old High German lazan, German lassen, Gothic letan "to leave, let"), from PIE *led-, extended form of root *‌‌lē- "to let go, slacken." If that derivation is correct, the etymological sense would be "let go through weariness, neglect."

"The shortening of the root vowel ... has not been satisfactorily explained" [OED]. Of blood, from late Old English. Other Old and Middle English senses include "regard as, consider; behave toward; allow to escape; pretend;" to let (someone) know and to let fly (arrows, etc.) preserve the otherwise obsolete sense of "to cause to." To let (someone) off "allow to go unpunished, excuse from service" is from 1814. To let on is from 1725 as "allow (something) to be known, betray one's knowledge of," 1822 as "pretend" (OED finds a similar use in the phrase never let it on him in a letter from 1637). To let out is late 12c. as "allow to depart" (transitive); intransitive use "be concluded," of schools, meetings, etc., is from 1888, considered by Century Dictionary (1895) to be "Rural, U.S." Of garments, etc., late 14c.

Let alone "abstain from interfering with" is in Old English; the phrase in the sense "not to mention, to say nothing of" is from 1812. To let (something) be "leave it alone" is from c. 1300; let it be "let it pass, leave it alone" is from early 14c. To let go is from c. 1300 as "allow to escape," 1520s as "cease to restrain," 1530s as "dismiss from one's thoughts." Let it go "let it pass, no matter" is as old as Chaucer's Wife of Bath: "But age allas Hath me biraft my beautee Lat it go, far wel, the deuel go ther with!" [c. 1395]. Let me see "show me" is from c. 1300.

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in (adj.)
"that is within, internal," 1590s, from in (adv.). Sense of "holding power" (the in party) first recorded c. 1600; that of "exclusive" (the in-crowd, an in-joke) is from 1907 (in-group); that of "stylish, fashionable" (the in thing) is from 1960.
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-in (1)
the adverb in attached to a verb as a word-forming element, by 1960, abstracted from sit-in, which is attested from 1941 in reference to protests and 1937 in reference to labor union actions (which probably was influenced by sit-down strike) but was popularized in reference to civil disobedience protests aimed at segregated lunch counters. As a word-forming element at first of other types of protests, extended by 1965 to any sort of communal gathering (such as love-in, 1967).
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-in (2)
word-forming element in chemistry, usually indicating a neutral substance, antibiotic, vitamin, or hormone; a modification and specialized use of -ine (2).
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in- (1)
Origin and meaning of in-
word-forming element meaning "not, opposite of, without" (also im-, il-, ir- by assimilation of -n- with following consonant, a tendency which began in later Latin), from Latin in- "not," cognate with Greek an-, Old English un-, all from PIE root *ne- "not."

In Old French and Middle English often en-, but most of these forms have not survived in Modern English, and the few that do (enemy, for instance) no longer are felt as negative. The rule of thumb in English has been to use in- with obviously Latin elements, un- with native or nativized ones.
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in (adv., prep.)

a Middle English merger of Old English in (prep.) "in, into, upon, on, at, among; about, during;" and Old English inne (adv.) "within, inside," from Proto-Germanic *in (source also of Old Frisian, Dutch, German, Gothic in, Old Norse i), from PIE root *en "in." The simpler form took on both senses in Middle English.

Sense distinction between in and on is from later Middle English, and nuances in use of in and at still distinguish British and American English (in school/at school). Sometimes in Middle English shortened to i.

The noun sense of "influence, access (to power or authorities)," as in have an in with, is first recorded 1929 in American English. to be in for it "certain to meet with something unpleasant" is from 1690s. To be in with "on friendly terms with" is from 1670s. Ins and outs "intricacies, complications of an action or course" is from 1660s. In-and-out (n.) "copulation" is attested from 1610s.

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in- (2)
Origin and meaning of in-

element meaning "into, in, on, upon" (also im-, il-, ir- by assimilation of -n- with following consonant), from Latin in- "in," from PIE root *en "in."

In Old French (and hence in Middle English) this often became en-, which in English had a strong tendency to revert to Latin in-, but not always, which accounts for pairs such as enquire/inquire. There was a native form, which in West Saxon usually appeared as on- (as in Old English onliehtan "to enlighten"), and some of those verbs survived into Middle English (such as inwrite "to inscribe"), but all now seem to be extinct.

Not related to in- (1) "not," which also was a common prefix in Latin, causing confusion: to the Romans impressus could mean "pressed" or "unpressed;" inaudire meant "to hear," but inauditus meant "unheard of;" in Late Latin investigabilis could mean "that may be searched into" or "that cannot be searched into." Latin invocatus was "uncalled, uninvited," but invocare was "to call, appeal to."

The trouble has continued in English; the hesitation over what is meant by inflammable being a commonly cited example. Implume (1610s) meant "to feather," but implumed (c. 1600) meant "unfeathered." Impliable can mean "capable of being implied" (1865) or "inflexible" (1734). Impartible in 17c. could mean "incapable of being divided" or "capable of being imparted." Impassionate can be "free from passion" or it can mean "strongly stirred by passion." Inanimate (adj.) is "lifeless," but Donne uses inanimate (v.) to mean "infuse with life or vigor." Irruption is "a breaking in," but irruptible is "unbreakable."

In addition to improve "use to one's profit," Middle English also had a verb improve meaning "to disprove" (15c.). To inculpate is "to accuse," but inculpable means "not culpable, free from blame." Infestive has meant "troublesome, annoying" (1560s, from infest) and "not festive" (1620s). In Middle English inflexible could mean "incapable of being bent" or "capable of being swayed or moved." In 17c., informed could mean "current in information," formed, animated," or "unformed, formless" ("This was an awkward use" [OED]). Inhabited has meant "dwelt in" (1560s) and "uninhabited" (1610s); inhabitable likewise has been used on opposite senses, a confusion that goes back to Late Latin.

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let-down (n.)
also letdown, "a disappointment," 1768, from let (v.) + down (adv.). The verbal phrase is from mid-12c. in a literal sense "cause to be lowered," of drawbridges, etc.; figuratively by 1754.
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