The Many Binary Operations of a Two Element SetDate: 11/03/2011 at 19:19:11 From: Shane Subject: binary operations Explain why the set S: = {a, b} allows 16 different binary operations. I understand (hopefully) the basic idea of a binary operation that is a map from S x S back to S. But I don't understand how a set of 2 elements can have 16 different binary operations. So far, I've only come up with four binary operations: a*b= ab = a + b = a - b = a/b Date: 11/03/2011 at 20:36:56 From: Doctor Peterson Subject: Re: binary operations Hi, Shane. A binary operation does not have to be one you know by name! In fact, addition, subtraction, multiplication, and division are defined on numbers, not on the letters a and b -- so they are entirely irrelevant. What we're looking at here is something much more abstract than the four familiar basic operations of arithmetic. You've got the definition: a binary operation on S is ANY mapping from SxS to S. If you were to list them, you would need to find all possible ways to assign values to each possible pair (x,y) where x and y are elements of S. So, first, how many such pairs are there? There are 2 ways to pick the first, and 2 ways to pick the second, for a total of 2*2 = 4 pairs. They are, in fact, (a,a), (a,b), (b,a), (b,b). Second, in how many different ways could you assign either a or b as the value of each of those? That will be the number of possible binary functions. We have 2 ways to assign a value to each of the 4 pairs; so there are 2*2*2*2 = 16 ways. Here's a slightly more concrete way to look at it. A binary operation * on S can be defined by filling in a table like this: * | a | b | --+---+---+ a | | | --+---+---+ b | | | --+---+---+ For example (entirely at random), here is one such operation: * | a | b | --+---+---+ a | b | b | --+---+---+ b | a | b | --+---+---+ Now, you can just make a list of all possible tables like this, since there are only 16; that would be a good exercise to get a feel for what an operation looks like. You'll also discover that some look sort of familiar, while others look very boring. But they are all valid binary operations. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 11/09/2011 at 20:21:22 From: Shane Subject: Thank you (binary operations) This is a belated thank you for your informative answer to my question. |
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