Binary Operations Explained in 5 Minutes

Binary Operations Explained in 5 Minutes

Binary Operations are all over mathematics. It's hard to avoid them. They're a fundamental concept.

Suppose we have a set of n objects S={x1,x2,x3,..,xn}. Then a binary operation takes two objects from the set and produces a new object, like this B(xi,xj)=z

The new object z may be in the set S or not. It could be a totally different object. When z is in the set we say that S is "closed" under the B operation. The point is that B operates on just two objects and this is why it's called a binary operation. Addition of numbers is an example of a binary operation. In this case order does not matter, so x+y=y+x, but this is not true in general and B(x,y) can be different than B(y,x). Binary operations for which B(x,y)=B(y,x) are known as Abelian operations after the Norwegian mathematician Niels Henrik Abel.

Can we generalize binary operations? Perhaps..

If can regard (xi,xj) as a sequence of two items from S then B is an operation that acts on sequences of length 2. So one way to generalize B would be to define an operation which acts on sequences of length, like this..


A is not a binary operation, it's an n-ary operation. Binary operations only act of sequences of length 2, but A acts on sequences of length n. This interpretation gets us to thinking about sequences. What exactly is a sequence?

A sequence is simply an ordered list of mathematical objects. The order is important. For example, (a,b) and (b,a) are different sequences. The other thing about a sequence is that, unlike a set, a sequence can contain duplicate objects. For example (a,a), (a,a,a), (a,a,b,b,b) and (a,a,a,...) are valid sequences. Also, it's easy to see that a sequence can be finite or infinite.

So now we can think of A as acting on a single object, a sequence of length n, and binary operations are the special case of n=2.

Let's summarize. We have a set S containing m objects. Then we define the "sequence set of S of order n" as the set of all sequences of length n made using the objects in S. Let's call this set Sn. How big is it? That's easy, it has m^n members. Sn is interesting because no matter what the size of S, Sn can be very big. For example, even if S has just one member, S={a}, the size of Sn is infinite. Our operator A operates on Sn.

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Content written and posted by Ken Abbott