Binary Operations

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Excerpt from the CSET study guide

TYPES OF BINARY OPERATIONS

Associative Operations    A binary operation is said to be associative if for all elements a, b and c we have (a x b) x c = a x (b x c). For convenience let’s drop the symbol for the operation and just write (ab)c = a(bc). The associative property then allows us to speak of abc without having to worry about whether we should find the answer to ab first and then that answer “multiplied” by c rather than evaluate bc first and then “multiply” a with that answer. Which ever way we process the expression we end up with the same element of the set. Note though that it does not say we can do the product in any order (i.e. ab and ba may not have the same value).

Sets that have an associative binary operation are known as semigroups. In many practical applications of studying binary operations on sets it is not unusual to discover they are associative but it is something that cannot be assumed. Indeed, one commonly known operation, the cross product on three dimensional real vectors is not associative. We should also guard against thinking that associativity implies identitities and inverses exist.

So why is it we need associativity to solve 3x=11 in the reals? Well, the inverse of 3 is 1/3 and so 1/3(3x) =11/3 and now we use associativity to rewrite this as (3/3) x = 11/3 etc.

Commutative Operations

A binary operation is commutative if ab = ba for ALL possible a and b in the set. Addition and multiplication in the reals are commutative operations whereas multiplication of matrices generally is not.

Note that the definition requires ab=ba for all pairs of elements. That some element commutes with all elements does not make the operation commutative.

Example

Let X = {a, b} and let F be the set of all functions from X to itself. There are four such functions and so |F| = 4. How each function maps elements of X is tabulated in the following table.

On F define a binary operation to be function composition. That is if u and v are functions then uv is the function

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