Intervals and Interval Notation

Intervals and Interval Notation The real number line is an infinite set of numbers that extends infinitely in both directions. There are many subsets of this infinite collection of numbers, some of which are infinite themselves. So let's just jump right in, this symbol, [2, 3.1]. The symbol [2,3.1] denotes an infinite set; but if the set has a finite bound—a maximum and a minimum—then it is also equal to the set of all real numbers X that satisfy two conditions: X must be greater than or equal to 2 and X must be less than or equal to 3.1. In other words, [2, 3.1] can be drawn on the real number line as follows. The closed interval [2, 3.1] contains the numbers 2, 3.1, and all numbers greater than or equal to 2 and less than or equal to 3.1. For example, 2.3 is in this closed interval because 2 is greater than or equal to 2.3 and also less than or equal to 3.1—and so are 3 and 3.1. One is not in this closed interval because it is not greater than or equal to two, even though it is less than or equal to three (which is inside the closed interval). Let's introduce the next - (5,8). Here, we 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: 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 endpoints. For example: 5.5 is in the open interval from 5-8; so is 5.0001; but sadly five is not in the open interval from 5-8 because it's 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 five and eight—those are right at either end; there are two extremes. For example, the set of all numbers in the open interval (-7.1, 15). You might already be able to guess this. On the open side of the inequality symbol, that means we use a strict inequality; on the other one, we use less than or equal to. This is an infinite set – all numbers in R such that -7.1 is strictly less than X is less than or equal to 15. So how am I drawing this? Here is an open -7.1; there is zero, just so we know where we are; and there's a closed 15 off the scale and everything in there. Let us take the other extreme. For example, [20,20.3). This will be the set of all X in R that are either less than 20 or equal to 20.3 but greater than -7.1. Again we'll draw my real number line: zero, 20, open interval from 20 to 20.3 and here's my point -7.1 to 15. Notice that in some sense, this interval seems small compared to the interval [-7.1, 15]. There are infinitely many numbers in this interval compared to the other one which had only two numbers: -7.1 and 15. We have seen closed intervals, such as [2,3.1]. We've seen open intervals such as (5,8). And within these are two species of half open intervals: for example, (2,3] and [20,20.3). Sometimes when we want to use fancy vocabulary, left open means the first of these half open intervals, right open means the second half open interval. That does not really matter. Let's take a look at another slightly more exotic notation. Suppose we write, close_two_comma_infinity. This just stands for the set of all X in R, such that X is greater than or equal to 2, full stop. You don't have to worry about negative infinity because every number is less than negative infinity. We often draw this on the real number line with zero here, a two and then going on forever up here. This is often what we call a ray or a half-line. You could also have, for example, minus_infinity_to_7.1_open which is the set of all X in R, that should be X is less than 7.1 and so on again. We're going to tie this example in with what we've already learned about algebra. Let's say someone asks you to solve for X if X+5=10 and X=5. You do some algebra and solve that X equals five, a number is an answer. On the other hand, suppose 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 of numbers; so let's do a little bit of algebra by subtracting five from all sides of this equation. Let's subtract five from the left side, I get -4 is less than or equal to, subtracting five from the middle yields just X. This tells me that X can be any number in the half open interval from [-4,5). In other words, as long as X is in this range then this statement is true. That concludes everything.