Etymology
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Words related to off

of (prep.)

Old English of, unstressed form of æf (prep., adv.) "away, away from," from Proto-Germanic *af (source also of Old Norse af, Old Frisian af, of "of," Dutch af "off, down," German ab "off, from, down"), from PIE root *apo- "off, away."

The primary sense in Old English still was "away," but it shifted in Middle English with use of the word to translate Latin de, ex, and especially Old French de, which had come to be the substitute for the genitive case. "Of shares with another word of the same length, as, the evil glory of being accessory to more crimes against grammar than any other." [Fowler]

Also by 1837 of in print could be a non-standard or dialectal representation of have as pronounced in unstressed positions (could of, must of, etc.)

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ab- 
word-forming element meaning "away, from, from off, down," denoting disjunction, separation, departure; from Latin ab (prep.) "off, away from" in reference to space or distance, also of time, from PIE root *apo- "off, away" (also the source of Greek apo "off, away from, from," Sanskrit apa "away from," Gothic af, English of, off; see apo-).

The Latin word also denoted "agency by; source, origin; relation to, in consequence of." Since classical times usually reduced to a- before -m-, -p-, or -v-; typically abs- before -c-, -q-, or -t-.
after (adv., prep.)

Old English æfter "behind; later in time" (adv.); "behind in place; later than in time; in pursuit, following with intent to overtake" (prep.), from of "off" (see off (adv.)) + -ter, a comparative suffix; thus the original meaning was "more away, farther off." Compare Old Norse eptir "after," Old Frisian efter, Dutch achter, Old High German aftar, Gothic aftra "behind;" also see aft. Cognate with Greek apotero "farther off," Old Persian apataram "further."

From c. 1300 as "in imitation of." As a conjunction, "subsequent to the time that," from late Old English. After hours "hours after regular working hours" is from 1814. Afterwit "wisdom that comes too late" is attested from c. 1500 but seems to have fallen from use. After you as an expression in yielding precedence is recorded by 1650.

*apo- 
also *ap-, Proto-Indo-European root meaning "off, away."

It forms all or part of: ab-; abaft; ablaut; aft; after; apanthropy; aperitif; aperture; apo-; apocalypse; apocryphal; Apollyon; apology; apoplexy; apostle; apostrophe; apothecary; apotheosis; awk; awkward; ebb; eftsoons; of; off; offal; overt.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit apa "away from," Avestan apa "away from," Greek apo "from, away from; after; in descent from," Latin ab "away from, from," Gothic af, Old English of "away from."
apo- 
before vowels ap-, word-forming element meaning "of, from, away from; separate, apart from, free from," from Greek apo "from, away from; after; in descent from," in compounds, "asunder, off; finishing, completing; back again," of time, "after," of origin, "sprung from, descended from; because of," from PIE root *apo- "off, away" (source also of Sanskrit apa "away from," Avestan apa "away from," Latin ab "away from, from," Gothic af, Old English of "away from," Modern English of, off).
back off (v.)
"retreat, stop annoying someone," by 1938, from the verbal phrase, from back (v.) + off (adv.).
beat off (v.)
"drive (something) away by violent blows," 1640s, from beat (v.) + off (adv.). Meaning "masturbate" is recorded by 1960s.
blast-off (n.)
"initial burst of energy that launches a rocket into space," 1950, from blast (v.) + off (adv.).
cast-off (adj.)

1709, "put aside, rejected," from verbal phrase cast off "discard, reject" (c. 1400), from cast (v.) + off (adv.). From 1741 as a noun, "person or thing abandoned as worthless or useless." Related" Cast-offs.

cut-off (n.)

also cutoff, 1640s, "act of cutting off," also "portion cut off," from verbal phrase cut off (see cut (v.) + off (adv.)). Sense of "new and shorter channel formed on a river" (especially the Mississippi) is from 1773; of road that cut off or shorten a route, from 1806; of clothing (adj.), from 1840. Cutoffs "jeans or other long pants trimmed down to be shorts" is by 1967.

The verbal phrase is attested from late 14c. as "detach by cutting;" from 1570s as "exclude from access" and "bring to an abrupt end;" and from 1590s as "intercept, stop the flow or passage of."