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Definition of undertake : 1. Hence, to guarantee; to promise; to affirm.
2. Specifically, to take upon one's self solemnly or expressly; to lay one's self under obligation, or to enter into stipulations, to perform or to execute; to covenant; to contract.
3. To assume, as a character.
4. To engage with; to attack.
5. To give a promise or guarantee; to be surety.
6. To have knowledge of; to hear.
7. To take or have the charge of.
8. To take upon one's self, or assume, any business, duty, or province.
9. To take upon one's self; to engage in; to enter upon; to take in hand; to begin to perform; to set about; to attempt.
10. To venture; to hazard.

Synonyms:

encounter, compress, squeeze, take over, take in charge, cut, adopt, get, acquire, shrink, sign on, obligation, accept, abbreviate, play, narrow, foreshorten, guarantee, take up, insure, warrant, reduce, press, admit, set to, set out, meet, launch, sign, abridge, concentrate, set about, try, sign up, pledge, institute, get off, enter, shorten, vouch, compact, start out, incur, get down, constrict, open, ensure, harness, assume, assure, kick off, essay, agree, secure, take on, tackle, condense, go about, lead off, take

develop, spring, bud, form, emerge, originate, dawn, commence, introduce, begin, induct, germinate, inaugurate, birth, stem, sprout, initialize, start, hatch, create, conceive, prepare, embark

chase, strive, hunt, tail, trail, aspire, dog, seek, stalk, track, search, intend, sleuth, follow, harry, quest

do, pursue, scheme, initiate, campaign

engage, busy oneself with, contract, commit oneself to, shoulder, hazard, proceed, commit, fall to, enter on or upon, go for in a big way, set in motion, volunteer, stake, perform, embark upon, go into, buckle to, take up for, move, exploit, offer, approach, pitch into, have in hand, go in for, devote oneself to, aim, answer for, have a try at, attempt, act, take upon oneself, assay, try out, promise, pledge oneself to, maneuver, fall into, launch into, venture, endeavor, take the plunge, address oneself to, launch forth

Usage examples:

• I do not undertake to say what the difference in the sensations is. - "A System Of Logic, Ratiocinative And Inductive (Vol. 1 of 2)", John Stuart Mill.
• Do you really think we ought to undertake this?" - "The Air Trust", George Allan England.
• I myself will undertake to form them into a company of foot- soldiers. - "The Nameless Castle", Maurus Jókai.