Sets Explained in 5 Minutes

Sets Explained in 5 Minutes

In mathematics a set is just a collection of objects. What type of objects? Any type. Of course you need a clear way to specify how an object belongs to a set.

Let's consider a simple example, the set containing the first 3 letters of the alphabet..

S={a,b,c}

Can we say anything mathematically interesting about this set?

Well, it's a finite set and contains 3 members. There's also a very clear rule to decide if an object belongs to the set. But we can do more, we can apply a mathematical technique to generate more structure. This technique is very simple, but turns out to be incredibly powerful. It's this..

Once you've defined something ask if contains things like itself. In this case we've defined a set S, so we ask if it contains any subsets. A subset of S is just another set made from the same objects. It's called a subset because you can think of it as contained inside S.

{a} is a subset, so is {a,c}, so is {b,c}

How many subsets does S have in total? In the count we'll include the null set { } which contains nothing and we'll also count the set itself {a,b,c} which contains everything. So here are all the subsets of S..

{ }
{a}
{b}
{c}
{a,b}
{a,c}
{b,c}
{a,b,c}

There are 8 in total, which just happens to be 2^3 where 3 is the number of objects in S. This is no coincidence. If our set contained n objects the number of subsets would be 2^n. This number gets big fast, for example..

A set with 26 members (such as the 26 letters of the alphabet) has 2^26=67108864 subsets. So from just 26 objects we can easily generate 67108864 new objects!

If you don't like the idea of counting { } and {a,b,c} as subsets then just say the number of subsets is (2^n)-2 and this makes almost no difference in the count. In the above example the number of subsets would be 67108862. Mathematicians call these subsets the proper subsets.

Content written and posted by Ken Abbott abbottsystems@gmail.com