How to check if some given definition is binary operations or not?

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Solution

For example If $* : N \times N - - N$ given by (a, b) --- 2a - b. check if this a binary operation.

If a = 2 and b = 4, then 2a - b = 2(2) - 4 = 0 Which doesn't belong to natural numbers. Therefore, it is not a binary operation.

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