What is the difference between 함께 and 같이?
They both mean together don't they?
How do you say with and without in Korean?

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for example, Let's go together : 같이 가자. Let's go separately : 따로 가자.

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같이 is more frequent used expressin to speak and 함께 is usually used in writing, in other words, a literary style. The meaning is Same.

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with= 와/과 같이/함께 without=이/가 없이 
for example: 
우리는 선생님과 같이 갔어요.(we went with our teacher) 
우리는 선생님이 없이 갔어요.(we went without our teacher)