So here's the real number line. We've talked before about finite sets, but in fact the real number line is an infinite set. Think of the large infinite number of things on this. Now there's a lot of subsets of this which are also ridiculously large and infinite. So let's just jump right in, this symbol, [2, 3.1]. This is an infinite set, but if an infinite set has got a finite bound -- we'll talk about how to describe what this is -- this is equal to the set of all numbers X in R, which satisfy two conditions. First, X has to be greater than or equal to two, and X has to be less than or equal to 3.1. In other words, to draw this on the real number line, here's two, here's 3.1, and X is trapped in between these two boundaries but it can hit up to both. So X is any number in the world as long as X is greater than or equal to two, less or equal to 3.1. So let's name some things in here. For example, 2.3 is in [2, 3.1] because two is less or equal to 2.3, and 2.3 is less or equal to 3.1. Three is in there, also 3.1 is in there, but one is not in the closed interval from two to 3.1, because one is not in fact less or equal to two even though it is actually less or equal to 3.1, well it's not greater or equal to two. OK, so that's a closed interval. Let's introduce the next idea. Suppose I give you (5,8) and here I use parentheses instead of those bracket symbols. This stands for an infinite set, the set of X in R, such that X is strictly greater than five and just strictly less than eight. So, the way we note this is, here is zero. We make a little open symbol for five, little open symbol for eight, then we just take all the things in between. So the idea is you have to be between five and eight but you can't hit up against those end points. So for example, 5.5 is in the open interval from 5-8, so is 5.0001 is in the open interval from 5-8 but sadly five is not in the open interval from 5-8, because five is not less than five even though it is less than eight. So you might want to think, by the way, what's the difference between the closed interval from 5-8 and the open interval from 5-8? They differ in exactly two numbers. They differ in five and eight. Five and eight are in the top one and not on the bottom. Okay so those are two extremes, here's two things in the middle, which we call "half-open intervals". Let's take for example, (-7.1,15]. You might already be able to guess this. On the open side, that means we use a strict inequality, and on the other one we use less than or equal to. This is an infinite set. Set of all X in R such that -7.1 is strictly less than X is less than or equal to 15. So how am I be drawing that, here is an open -7.1. There is zero, just so we know where we are, and there the closed 15 off the scale and all that stuff in there. Okay, we'll take the other extreme. So let's say for example, [20,20.3). This will be the set of all X in R plus the 20 is less three or equal to X and less than 20.3. Draw again the real number line. Here's my zero, let's see here's the 20 and here is an open 20.3. There's the stuff in here. But a little point to make, in some sense this seems really small, 20 to 20.3 is really tiny on the real number line compared to -7.1 to 15. It's got infinitely large number of numbers in there. That's the hilarious thing about the real number line. There's a lot of fancy math which you will get into behind it. We've seen closed intervals, like [2, 3.1]. We've seen open intervals, like (5,8). And within two species of half open intervals, like (2,3] and like [20,20.3). Sometimes if we want fancy vocabulary, the first of these half open interval is called left open, this is called right open, that doesn't really matter. OK, let's show you one more slightly more exotic form. Suppose we write, close_two_comma_infinity. This just stands for the set of all X in R, such that X is the greater than or equal to 2, full stop. You don't have to be less than infinity because every number is less than infinity. We often draw that on the real number line, here is zero, there's a two and we just take all the stuff here, kind of going on forever. This is often what we call a "ray". You could also have for example, a minus_infinity_ to_7.1_open. This is the set of all X in R, that should be X is less than 7.1, and you get the idea. Okay, let's close by tying in to the algebra we recalled in this video. We're already comfortable with the idea that if someone asked you to solve for X, and X+5=10, you do some algebra and you solve that X equals five. So X=5 is the answer, a number is an answer. Suppose on the other hand, someone gives you the following problem: Tell me everything you know about X if the following is true; 1_is_less_or_equal_to_X_plus_five_is_less_than_ten. So notice here, a single number is not the answer. For example, if X=4, then 4+5=9, nine is less than 10, nine is greater than or equal to one, but 3.9 would also work. In fact, it turns out that the answer is an interval. If we do a little bit of algebra, let's subtract five from all sides of this. Let's subtract five from the left side, I get -4 is less than or equal to, subtract five from the middle I just get plain old X that's why I subtracted five, that wasn't random. Subtract five from the right, I get 10 minus five or just five. In other words, the first puzzle tells me that the answer is any X in this range. So in other words, as long as X is in the half open interval from [-4,5), that's the answer. That tells me, any X in here tells me, that this up here is true. OK, that concludes everything.