Etymology
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Words related to deserve

de- 

active word-forming element in English and in many verbs inherited from French and Latin, from Latin de "down, down from, from, off; concerning" (see de), also used as a prefix in Latin, usually meaning "down, off, away, from among, down from," but also "down to the bottom, totally" hence "completely" (intensive or completive), which is its sense in many English words.

As a Latin prefix it also had the function of undoing or reversing a verb's action, and hence it came to be used as a pure privative — "not, do the opposite of, undo" — which is its primary function as a living prefix in English, as in defrost (1895), defuse (1943), de-escalate (1964), etc. In some cases, a reduced form of dis-.

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serve (v.)

late 12c., "to render habitual obedience to," also "minister, give aid, give help," from Old French servir "to do duty toward, show devotion to; set table, serve at table; offer, provide with," from Latin servire "be a servant, be in service, be enslaved;" figuratively "be devoted; be governed by; comply with; conform; flatter," originally "be a slave," related to servus "slave," which is of uncertain origin.

Perhaps from Etruscan (compare Etruscan proper names Servi, Serve, Latinized as Servius), but de Vaan says it is from Proto-Italic *serwo- "shepherd," *serwā- "observation," from PIE *seruo- "guardian" (source also of Avestan haraiti "heeds, protects"):

Rix 1994a argues that the original meaning of *serwo- probably was 'guard, shepherd', which underwent a pejorative development to 'slave' in Italy between 700 and 450 BC. Servire would be the direct derivative of servus, hence 'be a slave'; servare would in his view be derived from an older noun *serwa- or *serwom 'observation, heedance'.

By c. 1200 also as "to be in the service of, perform a service for; attend upon, be personal servant to; be a slave; owe allegiance to; officiate at Mass or other religious rites;" from early 13c. as "set food at table;" mid-14c. as "to wait on (customers)." From late 14c. as "treat (someone or something) in some fashion."

To serve (someone) right "to treat as he deserves" is recorded from 1580s. Sense of "be useful, be beneficial, be suitable for a purpose or function" is from early 14c.; that of "take the place or meet the needs of, be equal to the task" is from late 14c.; that of "suffice" is from mid-15c. Meaning "render active military service" is from 1510s. Sporting sense, in tennis, badminton, etc., first recorded 1580s. Legal sense "present" (a writ, warrant, etc.), "give legal notice of" is from early 15c.

He no schuld neuer wond
To seruen him fro fot to hond
["Amis and Amiloun," c. 1330]
deserved (adj.)

"rightfully earned, merited," 1550s, past-participle adjective from deserve (v.). Related: Deservedly (1540s).

deserving (adj.)

"that deserves," 1570s, present-participle adjective from deserve (v.). Related: Deservingly (1550s). Phrase deserving poor, those in need and unable to work through no fault of their own (the old, the sick, the lame, etc.) is by 1801.

undeserved (adj.)
late 14c., from un- (1) "not" + past participle of deserve (v.). Related: Undeservedly.
undeserving (adj.)
1540s, from un- (1) "not" + present participle of deserve (v.).