Does it mean that you no longer want someone, or that you have to leave someone, like it or not?
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It could mean that you are turning them in, to the police for example. It could also mean you are giving them up, like a bad habit.
To give someone up might apply to someone putting up their child for adoption, or something of a similar nature.
She gave the baby up because she wasn't ready to be a mother.
She wants to give the baby up because she's not ready to be a mother.
The term "to give someone up" is often used when a person lets someone move on with their life.
An example would be a love triangle where Person A loves Person B but Person B loves Person C instead.
In this case if Person A was to back off, give some space to B and C you could say that person say has given them up.
If "give up someone" is used in a "dating" situation, it means to let them go, or to stop dating them, etc.
In a different context, it means to "turn in" (e.g. a suspected criminal) to the authorities.