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Difference between Have a down on someone and Have an up on someone

have a down on someone—(also: have a downer on someone) have a grudge against smb.; dislike a person:

  • For some reason or other he had a down on me and I grew to hate him.

have an up on someonehave an advantage over a person:

  • I’m not bilingual though, so you do have an up on me.