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what are binary operations this is a vital concept in abstract algebra so thats what well be going over in todays wrath of math lesson try to give you a quick and effective explanation of what binary operations are well begin with our definition and then use some examples to clarify it so a binary operation is just a special type of function a binary operation on a set s is a function from the Cartesian product s cross s to that same set s this means that a binary operation takes two elements from the same set they could be the same element or they could be different and it maps them to exactly one element also in that set and you might not realize it but youre actually very familiar with many binary operations already for example consider a function G from the reals crossed with the reals to the reals defined like this G of a B is equal to a plus B addition on the real numbers is a binary operation it takes each pair of real numbers and maps it to exactly one other real number t