Etymology
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cross- 

word-forming element typically representing cross as a noun, adverb (cross-examine), adjective (crossbar), and in many words a confluence of them. "There is no distinct line of division between cross as an adjective and cross as a prefix. As a prefix, it often represents the adv. cross, or the prep. cross, across." [Century Dictionary]

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trans- 

word-forming element meaning "across, beyond, through, on the other side of, to go beyond," from Latin trans (prep.) "across, over, beyond," perhaps originally present participle of a verb *trare-, meaning "to cross," from PIE *tra-, variant of root *tere- (2) "cross over, pass through, overcome." In chemical use indicating "a compound in which two characteristic groups are situated on opposite sides of an axis of a molecule" [Flood].

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xylo- 
before vowels xyl-, word forming element meaning "wood," from Greek xylon "wood cut and ready for use, firewood, timber; piece of wood; stocks, a plank, beam, or bench," in New Testament, "the Cross," a word of uncertain origin. It seems to correspond with Lithuanian šulas "post, pole, stave," Russian šulo "garden-pole," Serbo-Croatian šulj "block," Old High German sul "style, pole," Gothic sauls "pillar," but the exact relationship is unclear, and Beekes asks, "Was the word taken from a non-IE substrate language?"
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