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binary compositionMy question is like that let S be a set of two elements. How many different binary composition can be defined on S? How many different commutative binary composition can be defined on S? The answers are 16 and 8, but I don't know how to solve it. In my opinion there can be only two binary composition. Please give a solution to this problem. |

If S = {x, y}, each of xx, xy, yx and yy may be x or y, so there are 16 possibilities. If the operation is commutative, each of xx, xy and yy may be x or y, but yx must equal xy, so there are only 8 possibilities. |

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