Interval And Set Notation


All right this video I'd like to talk about interval notation and set notation? Alright, so we are already familiar with say, you know, the notation, you know, X is less than 3 right? So we can graph it on a number line by simply saying, alright, here's 3, I'm. Sorry, I want to shade to the left.

All numbers are less than 3. So we shade a to the left put a little arrow didn't to note that it keeps going forever. And then we say is 3. The number 3 included are not included in the solution of this. Well, since X is. Less than 3 3 is not included.

And so we would use like an open circle or parenthesis from a previous video. So I'm going to go out and use a parenthesis all right so that's, not included this one look like such, but you could use also use an open circle. All right, as opposed to say this one where we have said, X is greater than 3 all right here's 3. And we want to shade out to the right this time because we want all numbers they're larger than 3 arrows on the end again. And this time since 3 is.

Included we would use like a closed circle, little dot or bracket, and I'm going to use a bracket, no closed bracket like that. Alright, so that would be the graphical representation of these two notations over here. Right?

The new notation we'll. Talk about is interval notation all right. So interval notation. We want to we want to literally write the interval that represents all the numbers that are solutions to a particular, you know, inequality like this one.

So over here, on the graph, this shaded stuff. That we do that's actually the interval. Alright, this is an interval all numbers in this shaded area here make this true. So to do the interval notation, we want to look for the smallest number well since this first one here, it goes out forever. He'll twist negative infinity. We say negative, infinity, being the, you know, the smallest number in our interval, even though you don't, even though you never get to infinity.

And then we have a comma and then what's the largest number in our interval. This case, You know, we get two three now we don't really get the three right because three is not included. We can't put the first number to the left of three because we can get infinitely close to three all right. So we do if we plug the two numbers for our interval, right?

The smallest to the largest to the smallest on the left largest on the right. And then we say all right it's. The number is the number three included or not included in the solution, well, it's not included. So just like we did on. The graph here, we're going to have a parenthesis and then infinity, whether it's positive or negative. It is never included. So it will always be a parenthesis all right will always be a print to see when we're when we are messing with infinity and then down here on this.

Second one here we got the smallest number is three, you know over here. And then it goes out to positive infinity, positive thing on the right, right? This would be the interval. If since three is included, we put a bracket just like we did.

On the number line and then infinity is not included. So we would put a parenthesis. And these two things are here. This represents the interval of this graph.

And this particular algebraic notation over here on the left. Alright. So this is called interval notation. There are some variations of it. So for this next one let's say, we've got this expression. Negative 2 is less than or equal to X is less than 5.

Alright, so we've got negative 2 on our number line. We can plot and 5 on our number line that. We can plot, alright? And this is saying that X the values we're looking for have to be greater than negative 2. But at the same time, they have to be less than 5. Everybody sees that that's, really what this compact expression is same really another way to put that in English is all numbers that are between negative 2, & 5, right so greater than negative 2. But at the same time, less than 5 would be all numbers between negative 2, & 5, and we would just shade that part right there.

And then we would. Determine the endpoints of the interval, for example, negative 2 here is included because it's equal to. So we would have a bracket and 5 is not included. So we would have a print right? And then interval notation would look just like that.

You have a bracket negative. Two commas, five parentheses, right. Interval notation. One more I want to do an interval notation that will briefly go through the set notation all right. So, what if we have it looking like this? This is called a calm these.

Last two are called. Compound inequalities, you'll see somewhere down the road. So here, we're, looking for all numbers that are solutions to this.

First inequality, X is less than or equal negative two or any numbers that are solutions to the second inequality X is greater than five all right. So again, we've got the negative two and five that we can plot on our number line. And then we look at all right since it's, an, or we're, looking for all the numbers that make either one of these true all right so X is less than or.

Equal to negative two, well, that'd mean, we'd shade out to the left here less than negative two. And since negative two is included, you'd have a bracket like such. And then for the second one all numbers are greater than fives, which shade out to the right here. Five put a little arrow on say, it goes on forever. And then five is not includes. We'd put a parenthesis all right so that's.

What the graphical representation would look like for an algebraic solution, such as this with an or typically with an or. Interval notation would look very similar. We've got two intervals though right, as opposed to these first three examples, we had one interval now we have two, so we just do them both so this first interval would have negative infinity to negative two included. Then we would have the second interval front to see five to infinity. And what we do is we use. This is you in between them like this, a without a tail, I call it for the word Union and really that's the mathematical version of the.

Word or right, so the would this down here would be read any number in this interval infinity to negative two. So any number less than negative two or what the Union means any number greater than five all right. So this is what the interval notation would look like, which is a little different. You know, so whenever you want to join intervals with an, or then we use this you without a tail, all right, okay? So that's interval notation set notation is uses the squiggly braces all right and. It's, whatever letter you're playing with.

So in this case, we're playing with X and then there's this vertical bar. Now that vertical bar means such that all right so write that down. This vertical bar means such that. So we read this at the moment as at all X's such that, well, such that what happens? Well, back here on the algebraic expression, such that X is less than three. So this right here is called to set notation all right. We read all X's such that X is less than three, which means that any number.

Less than three is are the numbers that we want for this for the solution right so say, my dad down here, we'd have all X's such that X is greater than or equal to three for the third one down here. We'd have all X's such that X is between negative two and five right. And then for this last one, we would have all X's such that X is less than negative 2 or X is greater than five. So if you already have the algebraic solution, then writing, an in set notation is pretty easy just throw in little X, such. That or I should say, whatever variable you're playing with, it might not be X that you're playing with the new thing. Here is really the interval notation.

And you should know how to use all four of these notations and be able to go between all four of these notations. So if you see X is less than three, we know, hey, that means negative Finite, comma, three and interval notation, which also means, hey, all X's, such that X is less than three in set notation. You want to be able to go between all four of them. Fairly easily all right? That's.

It studies. Well, please, let me know if you have any questions.

Dated : 14-Apr-2022

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