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in sporting use, a variant of by (prep). Originally in cricket, "a run scored on a ball that is missed by the wicket-keeper" (1746); later, in other sports, "position of one who is left without a competitor when the rest have drawn pairs" (1883), originally in lawn-tennis.
shortened form of good-bye. Reduplication bye-bye is recorded from 1709, though as a sound used to lull a child to sleep it is attested from 1630s.