Etymology
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Words related to syn-

asymptote (n.)

"straight line continually approaching but never meeting a curve," 1650s, from Greek asymptotos "not falling together," from a- "not" (see a- (3)) + assimilated form of syn "with" (see syn-) + ptotos "fallen," verbal adjective from piptein "to fall," from PIE root *pet- "to rush; to fly." Related: Asymptosy.

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asyndeton (n.)

"figure of speech consisting of omission of conjunctions," 1580s, from Latin, from Greek asyndeton, neuter of asyndetos "unconnected," from a- "not, without" (see a- (3)) + syndetos, from syndein "to bind together," from syn- "together" (see syn-) + dein "to bind," related to desmos "band," from PIE root *dē- "to bind" (see diadem).

"I pitied thee, Took pains to make thee speak, taught thee each hour One thing or other." ["The Tempest"]
idiosyncrasy (n.)
c. 1600, from French idiosyncrasie, from Latinized form of Greek idiosynkrasia "a peculiar temperament," from idios "one's own" (see idiom) + synkrasis "temperament, mixture of personal characteristics," from syn "together" (see syn-) + krasis "mixture," from PIE root *kere- "to mix, confuse; cook" (see rare (adj.2)).

Originally in English a medical term meaning "physical constitution of an individual;" mental sense "peculiar mixture" of the elements in one person that makes up his character and personality first attested 1660s. In modern use, loosely, one's whims, habits, fads, or tastes. Sometimes confused in spelling with words in -cracy, but it is from krasis not kratos.
sanhedrim (n.)
1580s, from Late Hebrew sanhedrin (gedola) "(great) council," from Greek synedrion "assembly, council," literally "sitting together," from syn- "together" (see syn-) + hedra "seat," from PIE root *sed- (1) "to sit." Abolished at the destruction of Jerusalem, C.E. 70. The proper form is sanhedrin; the error began as a false correction when the Greek word was taken into Mishanic Hebrew, where -in is a form of the plural suffix of which -im is the more exact form.
sejm (n.)

congress of the Polish republic, 1690s, from Polish sejm "assembly," from *syn-imu, literally "a taking together," from *syn- "together" (see syn-) + PIE root *em- "to take."

sy- 
form of syn- before -s- or -z-.
syl- 
assimilated form of Greek syn- before -l-.
syllable (n.)

late 14c., from Anglo-French sillable, alteration of Old French silabe "syllable" (12c., Modern French syllabe), from Latin syllaba, from Greek syllabē "that which is held together; a syllable, several sounds or letters taken together," i.e. "a taking together" of letters; from syllambanein "take or put together, collect, gather," from assimilated form of syn- "together" (see syn-) + stem of lambanein "to take" (see lemma). The unetymological -le apparently is by analogy with participle and principle.

syllepsis (n.)
use of a word at once in both a literal and metaphoric sense, 1570s, from Late Latin syllepsis, from assimilated form of Greek syn "together" (see syn-) + lepsis "a taking," related to lambanein "to take" (see lemma). Related: Sylleptic.
syllogism (n.)
late 14c., from Old French silogisme "a syllogism, scholastic argument based on a formula or proof" (13c., Modern French syllogisme), from Latin syllogismus, from Greek syllogismos "a syllogism," originally "inference, conclusion; computation, calculation," from syllogizesthai "bring together before the mind, compute, conclude," literally "think together," from assimilated form of syn- "together" (see syn-) + logizesthai "to reason, count," from logos "a reckoning, reason" (see Logos).