Up until now, everything I’ve talked about has been quite concrete. It’s been about certain types of numbers (so a number either ‘is’ a certain type of number, or it ‘is not’) or about actually solving things / showing things are true. In maths, a ‘set’ is a little bit more abstract than this – though not much, so don’t panic yet.
Sometimes, we refer to a collection of things as a set. So, for example, you may have a set of coins, a set of cups, a set of stickers, etc. If you have a set of something, it pretty much means you have a collection of things that all share something in common. We can talk about a mathematical set in a similar way.
In fact, we’ve already seen some sets! Back in the ‘Different Types of Numbers’ entry, we saw loads of different sets. The natural numbers (all the whole numbers that are positive) are a set of numbers, the integers (all whole numbers) are a set, the ‘real numbers’ are a set and so forth. These are some of the more basic, yet also more important, sets that we have in maths.
But we can create a set of numbers in any way really. We could have “the set of all square numbers” (1, 4, 9, 16, 25 and so on). We could have “the set of real numbers that are between 0 and 1” (so things like $ latex \frac{1}{2}$, 4 latex \frac{2}{3}$, etc.). Use your imagination and you can probably think of any sort of set of numbers.
So what’s the use of sets? Why do we care if we have sets of numbers? Well, there’s many reasons. Sometimes, certain equations will only work on certain sets of numbers. For example, if you look at $ latex \sqrt(x)$ and you want it to give you a real number, x has to be in “the set of all positive real numbers”. If you wanted it to be a whole number, x would have to be in “the set of all square numbers”. In other words, it lets us restrict other things so that they always work.
We can also think of sets being ‘contained’ in other sets. So, if we have, say, “the set of numbers between 1 and 2”, that is contained in “the set of real numbers”. This can be really useful. For example, if we want to see if something works for numbers between 1 and 2, it might actually be easier to show that it works for all real numbers (which would then mean it works for all numbers between 1 and 2). Sounds a bit weird, but sometimes Maths does!
There’s a lot more exciting things that mathematicians can do with sets. They can be studied in their own right, and have lots more interesting properties that can be gone into. But for my first aim of trying describe my summer project, I won’t go into them now. Maybe another time…
Josh Explains Maths 