# Binary Operations : Cayley Table

Consider the Cayley table: (see file)

Show that there is only one way to complete table (1) so that the resulting operation is associative, and that the result makes {a,b} into a commutative monoid.

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Proof. Denote the binary operation by *. By the hypothesis, we know that a*b=b and b*b=a. ...

#### Solution Summary

A Cayley table is completed so as to satisfy set conditions. The associative operation resulting from a commutative monoid is determined.

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