Lecture Note
University
Duke UniversityCourse
Data Science Math SkillsPages
3
Academic year
2023
KatrCrayon
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31
Intervals and Interval Notation The real number line is an infinite set of numbers that extends infinitely in bothdirections. There are many subsets of this infinite collection of numbers, some ofwhich are infinite themselves. So let's just jump right in, this symbol, [2, 3.1]. The symbol [2,3.1] denotes an infiniteset; but if the set has a finite bound—a maximum and a minimum—then it is alsoequal to the set of all real numbers X that satisfy two conditions: X must be greaterthan 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] containsthe numbers 2, 3.1, and all numbers greater than or equal to 2 and less than orequal to 3.1. For example, 2.3 is in this closed interval because 2 is greater than orequal to 2.3 and also less than or equal to 3.1—and so are 3 and 3.1. One is not inthis closed interval because it is not greater than or equal to two, even though it isless 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 bracketsymbols. This stands for an infinite set, the set of X in R, such that X is strictlygreater than five and just strictly less than eight. So the way we note this is: zero; wemake a little open symbol for five; little open symbol for eight; then we just take allthe things in between. So the idea is you have to be between five and eight but youcan't hit up against those endpoints. For example: 5.5 is in the open interval from5-8; so is 5.0001; but sadly five is not in the open interval from 5-8 because it's notless than five even though it is less than eight. So you might want to think: by theway, what's the difference between the closed interval from 5-8 and the open intervalfrom 5-8? They differ in five and eight—those are right at either end; there are twoextremes.
For example, the set of all numbers in the open interval (-7.1, 15). You might alreadybe able to guess this. On the open side of the inequality symbol, that means we usea strict inequality; on the other one, we use less than or equal to. This is an infiniteset – 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 knowwhere 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 inR that are either less than 20 or equal to 20.3 but greater than -7.1. Again we'll drawmy real number line: zero, 20, open interval from 20 to 20.3 and here's my point -7.1to 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 otherone 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 thefirst 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 isgreater than or equal to 2, full stop. You don't have to worry about negative infinitybecause every number is less than negative infinity. We often draw this on the realnumber line with zero here, a two and then going on forever up here. This is oftenwhat 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 lessthan 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 algebraand solve that X equals five, a number is an answer. On the other hand, supposesomeone gives you the following problem: Tell me everything you know about X ifthe following is true; 1_is_less_or_equal_to_X_plus_five_is_less_than_ten. Sonotice here, a single number is not the answer; for example, if X=4 then 4+5=9, nineis less than 10, nine is greater than or equal to one but 3.9 would also work. In fact, itturns out that the answer is an interval of numbers; so let's do a little bit of algebra bysubtracting 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 fivefrom the middle yields just X. This tells me that X can be any number in the half openinterval from [-4,5). In other words, as long as X is in this range then this statement istrue. That concludes everything.
Intervals and Interval Notation
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