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# Intervals and interval notation

## Video transcript

what I hope to do in this video is get familiar with the notion of an interval and also think about ways that we can show an interval or interval notation so right over here I have a number line now let's say I wanted to talk about the interval on the number line that goes from let's say from negative 3 to 2 so I care about this let me just in a different color let's say I care about this interval right over here so I care about all the numbers from negative 3 negative 3 to 2 so in order to be more precise I have to be clear am I including negative 3 & 2 or am I not including negative 3 & 2 or maybe I'm just including one of them so if I'm including negative 3 & 2 then I would fill them in so this right over here I'm filling it negative 3 & 2 in which means that negative 3 & 2 are part of this interval and when you include the endpoints this is called a closed interval closed interval closed interval and I just showed you how I can depict it on a number line by actually filling in filling in the endpoints and there's multiple ways to talk about this interval mathematically I could say that this is all of the let's say that this is let's say this number line is showing different values for X I could say these are all of the X's that are between negative 3 negative 3 & 2 & 2 and notice I have negative 3 is less than or equal to X so that's telling us that X could be equal to that X could be equal to negative 3 and then we have X is less than or equal to positive 2 so that means that X could be equal to positive 2 so it is a closed interval now another way that we could depict this closed interval is we could say okay we're talking about the interval between and we could use brackets because it's a closed interval negative 3 & 2 and once again I'm using brackets here these brackets tell us that we include so this bracket on the Left say that we include negative 3 and this bracket on the right says that we include positive - in our interval now sometimes you might see things written a little bit more math you might see something like that like X is a member of the real numbers such that so and I could put these curly brackets around like this these curly brackets say that we're talking about a set of values and we're saying the set of all X's that are a member of the real number so this is just fancy math notation it's a member of the real numbers I'm using the Greek letter epsilon right over here to say member of the real numbers such that this little this this vertical line here means such that negative 3 is less than X is less than negative 3 is less than or equal to X is less than or equal to 2 I could also write it this way I could write X is a member of the real numbers such that X is a member such that X is a member of this closed said I'm including I including the endpoints here so these are all different ways of of denoting or depicting the same interval let's do some more examples here so let's let me draw a number line again so number line and now let me do let me just do an open interval an open interval just so that we see we clearly can see the difference so let's say that I want to talk about the values between negative 1 & 4 negative let me do this in a different color so the values between negative 1 & 4 but I don't want to include negative 1 & 4 so this is going to be an open interval so I'm not going to include 4 and I'm not going to include and I'm not going to include negative 1 notice I have open circles here over here I had closed circles the closed circles told me that I included negative 3 & 2 now I have open circles here so that says that I'm not it's all the values in between negative 1 & 4 so negative 1 or negative point 9 9 9 9 9 9 9 is going to be included but negative 1 is not going to be included and 3 point 9 9 9 9 9 9 9 is going to be included but 4 is not going to be included so how would we what will be the notation for this well here we could say X is going to be a member of the real numbers such that negative one I'm not going to say less than or equal to because X isn't X can't be equal to negative one so negative one is strictly less than X is strictly less than four notice not less than or equal because I can't be equal to four four is not included so that's one way to say it or another way I could write it like this X is a member of the real numbers such that X is a member of now the interval is from negative one to four but I'm not going to use these brackets these brackets say hey let me include the endpoints but I'm not going to include them so I'm going to put the parenthesis right over here the parenthesis so this tells us that we're dealing with an open interval so this right over here let me make it clear this is an open an open interval now you're probably wondering okay in this in this case both endpoints were included is the closed interval in this case both at endpoints were excluded it's an open interval can you have things that have one endpoint included and one an endpoint excluded and the answer is absolutely let's see an example of that so I'll get another number line here another another number line and let's say that we want to and actually let me do it the other way around let me write it first and then I'll graph it so let's say we're thinking about all of the X's that are a member of the real numbers such that let's say negative 4 let's say negative 4 is not included is less than X is less than or equal to negative 1 so now negative 1 is included so we're not going to include negative 4 negative 4 is strictly less than not less than or equal to so X can't be equal to negative 4 so open circle there but X could be equal to negative 1 has to be less than or equal to negative 1 so it could be equal to negative 1 so I'm going to fill that in right over there and then it's everything in between if I want to write it with this notation I could write X is a member of the real numbers such that X is a member of the interval so it's going to go between negative 4 and negative but we're not including negative for we have an open circle here so I'm going to put a parenthesis on that side but we are including we are including negative 1 we are including negative 1 so we put a bracket on that side and so that would be that right over there would be the notation now there's other things that you could do with interval notation you could say well hey everything except for some value so let me give another example let me like let's get another example here let's let's say that we we want to talk about all the real numbers except for 1 so we want to include so we want to include all of the real numbers all of the real numbers except except for one except for 1 so we're going to exclude one right over here so open circle but it can be any other any other real number so how would we how would we how would we denote this well we could write we could write X is a member of the real numbers such that X does not equal does not equal 1 so here I'm saying look X can be a member of the real numbers but X cannot be equal to 1 it can be anything else but it cannot be equal to 1 and there's other ways of denoting this exact same interval you could say you could say X is a member of the real numbers such that X is you could say such that X is less than 1 or X is greater than 1 so you could you could write it just like that or you could do something interesting I this is probably this is the one that I would use this is the shortest and it makes it very clear say hey everything except for one but you can even do something fancy like you could say X is a member of the real numbers such that X is a member of the set going from negative infinity to 1 not including 1 or X is a member of the set going from or a member of the interval going from 1 not including 1 all the way to positive all the way to positive infinity and when you're talking about a negative infinity or positive infinity you always put a you always put a parentheses and the view there is you can never include everything all the way up to infinity it's it's it needs to be at least open at that endpoint because infinity just keeps going on and on so you always want to put a parenthesis if you're talking about infinity or negative infinity it's not really an endpoint it keeps going on and on forever so you use the notation for open interval at least at that at that end and notice we're not including we're not including one either so if X is a member of this interval or that interval it essentially can be anything other than one but this would have been the simplest notation to to describe that