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Given an example of a set $G$ and an operation $*$ on $G$ such that $*$ is not a binary operation on $G$ but associative, identity and inverses properties hold?

Basically, try to find an example to show that the closure property must be hold to be a group

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How about the set $G = \{-1,0,1\}$ under usual addition?

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All integers under addition. Let Z= all integers: where G= (Z,+,0) Closure property:

consider the subset {1,2} under integers, where 1+2=3 and 3 is an integer. Thus, proves the closure property holds for all integers under addition.

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What's the inverse of 2? – Gamma Function Jun 3 '14 at 0:31
    
the inverse of 2 is -2 – April Jun 3 '14 at 13:06

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