Sets

A set is a collection of things. That's right, it's that simple. What those things are, depends on what you're talking about. For example, you can have a set of fractions, or a set of numbers, or a set of integrals, or a set of whatever (Sometimes mathematicians try to be cool and talk about a set of unicorns, or pink elephants, but that's just stupid.)

Sets are denoted by a set of curly brackets, like this: { }. And you put the things in your set inside the brackets, so that a set of two fractions would look like this.



Sometimes you might want to give it a name, (but not an imaginative one, because we're mathematicians) like A, so that you would write



That's all fine and dandy if you can write down the list of things you want to put in you set. But what if you want to talk about a whole bunch of things, where's there's a lot of them. Say an infinite number of them. Like, all the fractions between 1 and 3. Obviously there 3/2, and 4/5, and 8/3. But there's a whole lot of them. How can we describe such a set?

Well, we write down inside the curly brackets the rules for what's in the set and what's not. (This is excitingly close to the important concept of the Axiom of Choice (what a great name; "The Axiom Of Choice"), but I'm not going to talk about it, because I'm not an expert, and it deserves it's own blog, as it was the sort of idea and revelation that shook the very foundations of mathematics. (If we really want to know, it's about whether you can even talk about an infinite number of objects or not. Or something like that))

So, getting back to it, we write down a rule for what's in the set, and hope that someone else knows what we're talking about it. For example, we might want to talk about all of the numbers between 1 and 3 (There should be an infinite number of them, so we'd better make a rule!). We would write this like



One would read this out loud (if one was a mathematician) as

A is the set of all things which are bigger than 1 and less than 3.

The colon is useful there when you want to be more specific about what sort of objects you're applying the rule to. For example, if you wanted to talk about all the fractions between 1 and 3, where is the set of all fractions, you'd write



where, as you can probably guess, the symbol means `is one of', or in maths speak, `is in'. In other words

A is the set of all numbers that are fractions () and are bigger than 1 and less than 3 ( ).