A group is defined as a set with a binary operation define on it, which satisfies closure, associative, identity and inverse properties. Why is closure property included in the definition what is the significance of this property?
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To guarantee that the product of two elements remains in the group. – Michael Albanese Jun 18 '14 at 09:47
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1If you did not have closure, where would this operation "live"? In fact, it is not an operation at all if it does not satisfy closure, which is why this is often left out in the definition of a group. – Tobias Kildetoft Jun 18 '14 at 09:50
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Pardon me if this is a stupid question, isn't an operation same as a function that can be define from one set to another set. Cant we say that this binary operation is a function acting on two input elements from a set and the result will be in another set. – sai prasad Jun 18 '14 at 09:52
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2We could, but it just happens that we do not. A binary operation os a set $X$ is by definition a map from $X\times X$ to $X$. – Tobias Kildetoft Jun 18 '14 at 09:56
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Thanks Tobias! Can you explain what is the role of associative property. – sai prasad Jun 18 '14 at 10:11