Bye

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mid 16th century (denoting a side issue or incidental matter): from by.

wiktionary

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Variant form of by, from Old English bī(being near).

Shortened form of goodbye.

Alternative forms.

etymonline

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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.