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A non-empty set  together with at least one binary operation defined on it is called an algebraic structure. Thus if  is a non-empty set and “ ” is a binary operation on , then  is an algebraic structure.
are all algebraic structures. Since addition and multiplication are both binary operations on the set  of real numbers,  is an algebraic structure equipped with two operations.
Example: If the binary operation  on the set of rational numbers is defined by
 for every 
show that  is commutative and associative.
Solution:
(1) “ ” is commutative in  because if , then
(2) “ ” is associative in because if  then
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