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The Many Binary Operations of a Two Element Set

Date: 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 

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.
Associated Topics:
College Modern Algebra

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