Etymology
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bye (n.)

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" (1868).

bye (interj.)

1709, shortened form of good-bye. Also compare lullaby. Reduplication bye-bye is recorded from 1736; as a sound used to lull a child to sleep it is attested from c. 1500.

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Definitions of bye

bye (n.)
an automatic advance to the next round in a tournament without playing an opponent;
he had a bye in the first round
Synonyms: pass
bye (n.)
a farewell remark;
they said their good-byes
Synonyms: adieu / adios / arrivederci / auf wiedersehen / au revoir / bye-bye / cheerio / good-by / goodby / good-bye / goodbye / good day / sayonara / so long
From wordnet.princeton.edu