Finite Math: Set Operations and Notation

hello welcome to the next video in my series on basic finite mathematics now a few things before we get started number one if you are a student watching this video and you are struggling and I class right now I want you to stay positive and keep your head up think of all that you have accomplished to get this far so you know you have it in you and I know the right amount of hard work practice and patience you can get through it I have faith in you many other people around you have faith in you therefore socha Jew number 2 please feel free to follow me here on YouTube and or on Twitter that way when to upload a new video you know about it another topic of the video if you liked it please give it a thumbs up share it with colleagues or classmates are added to a playlist but that does encourage me to keep making them on the flip side if you think there is something I can do better please leave the constructive comment below the video and I will try to incorporate those ideas into future ones and finally just keep in mind these videos are meant for people who are relatively new or brand-new to finite mathematics so I will just be going over the basic concepts and I will be doing so in a very slow deliberate manner not only do I want you to understand what's going on but also why so all that being said let's go ahead and get started so this video is sort of an introductory video to the idea of sets and counting more specifically in later videos we're going to talk about the dreaded by some people Venn diagrams but before we can talk about Venn diagrams and intersections of sets and things like that we have to first understand exactly what sets our set operations and how we notate sets because to do the more complex work you have to have this foundational understanding so let's go ahead and get started doing that so what is a set exactly well you know I try to give my students everyday concrete examples so the easiest way to think about a set are you know things and everyday life so think of a set of golf clubs so a full set of golf clubs has a driver club in it it has a 7-iron in it it has a putter in it usually so a set of golf clubs is a bunch of different other clubs which we would call elements and we'll talk about that in a minute so a set of golf clubs or a set of dining dishes so here we have three plates and a cup and they come in a set from the store or a set of wizards I put that in there for fun but it can be treated to a set of lizards so think of things which are grouped together based on some criteria that makes them similar so if we go buy a set of golf clubs at the sporting goods store we're not going to open up the box and find a bag of dog food likewise if we go to the store and we buy a set of dishes in the box and we open up the box we're not going to find a pair of bedroom slippers because that obviously does not go with the set most likely and of course in a Harry Potter movie where we have a set of wizards we don't expect we don't expect to see a set of Ewoks from Star Wars walk through the movie so a set is just a group of similar things based on some criteria and that's the easiest way to think about it now of course in mathematics classes we have more formal definitions okay so let's go ahead and look at these graphically and in terms of how we notate them so we're going to have a set here which we're going to call set in and inside this set it's kind of like a fence a found these numbers and inside this set in we have the numbers one through nine so one two three four five six seven eight nine the individual things inside the set are called elements so one is an element of set in six is an element of set in four is an element etc so the individual things inside the set are called elements now we write that in this manner so set in equals and then in braces we'd list the elements so one two three four five six seven eight nine now this symbol is one that throws students for a loop because unfortunately sets and counting and set notation is often skipped in many high schools so my students always like I've seen integral calculus but I've never seen this symbol well the good thing is this symbol is very simple all it means is is an element of so for example four is an element of set in because there it is right there in the middle of our set now there are sort of an opposite symbol that looks like this and it means not an element of so for example the number 12 is not an element of set in so basically these symbols just tell us whether or not individual elements are members of a set or not now let's talk about equivalent sets so we have our same set in over here on the left now we're going to have a new set M over there on the right so in set in we have the same nine or the same numbers 1 2 3 4 5 6 7 8 9 over here on set M we have the same thing 1 2 3 4 5 6 7 8 9 so these are equal to each other they are equivalent so M equals N and N equals M because mmm have the exact same elements so if we list the elements and set N 1 2 3 4 5 6 7 8 9 if we list the elements in M we have 1 2 3 4 5 6 7 8 9 they are equivalent sets now it's important to note here that the order in which we list the elements does not matter so I could list in as it is but I could list em as 9 8 7 6 5 4 3 2 1 it doesn't matter the order does not matter in terms of how you list the elements let's talk about subsets now I just want to point out here there is sort of this confusing part of subsets that I'll have to explain in a later slide so just hang with me for subsets we'll talk about proper subsets and empty sets in the next couple of slides and then sort of a summary slide it will all sort of come together so if it seems confusing now just just hang with me so we have our subset or said I'm sorry over here in set in our same numbers 1 2 3 4 5 6 7 8 9 now over here we have set e which is 2 4 6 & 8 now notice that everything that is in E is also in n so 2 is in both of them 4 is on both of them 6 is in both of them and 8 is on both of them now when we go ahead and learn how to draw Venn diagrams there is a graphical way of representing this however since we're just learning about sets in general we'll skip over that for now but E is a subset of n because everything in E is also an N now the symbol we use for a subset of sort of this C shape with a line underneath it now what we're saying here is that e can be any subset of n including the equivalent step and this is where it gets kind of weird now obviously set and said e are not equivalent but said E is a subset of said in but also if you remember in the previous slide we had set in and set em and they were equivalent and in terms of subsets they are subsets of each other so you can think of it as a twin so here are sets here and an equivalent set is a subset of its twin and again this is where it can get kind of confusing but hang with me there's a slide at the end that kind of summarizes all this and an easy-to-use table so e can be any subset of n including its twin is also including its empty set but we haven't talked about empty sets yet but we will here in a second now let's talk about proper subsets so we have our set in over here the same thing we've been doing and we have set e over here on the right it's the same two sets we had in the previous slide remember we said that E is a subset of n which is true now E is also a proper subset of n so we have subsets and proper subsets well what's the difference well in terms of our nomenclature and our symbols we use this symbol so it's kind of the C shape but there's no line under it and that's there's an important reason why that is now it can be any subset of in including the empty set so you would qualify here but e cannot be equivalent to n and that's the primary difference between subsets and proper subsets for all practical purposes sets and subsets are the same thing except when it comes to proper subsets the equivalent sets don't count so equivalent subsets are not proper subsets so again we'll look at that here in a second and it should make a bit more sense than me trying to stay set you know a thousand times now empty sets so here we have our set in again and our numbers in there and over on the side we have another set Z but it is empty there is nothing in it so we call that an empty set so it's often written with this symbol down here below now that's not 0 okay that sort of did annul that symbol sort of means null it means empty so it's kind of a zero shape with a line through it sometimes it's just two braces with nothing in there sometimes written like that but that's an empty set so Z remember the symbol Z is a subset of M so the empty set the empty set is a subset of n it's also a proper subset of n remember they're not equivalent so the proper thing doesn't get in the way so the empty set is both a proper and a basic subset of any set now every every set does have an empty set as a subset so whatever set we have over here we have one two three four five six seven eight nine of course an empty set is a subset of n we over there we could have you know all the oceans on the world but we can still have an empty set that is none that is none of them so Z is a subset of any set and actually have every set okay so here is going to be a table that will hopefully bring all this together and a nice easy to read format so let's say we have a set J and it contains the numbers 0 1 & 2 now we can make many subsets out of those numbers so we'll call that subset set K so set K can be empty set K could just have the number 0 set K could have those number 1 set K could have just the number 2 set K could be the numbers 0 and 1 set K could be 0 and 2 set K could be 1 and 2 and set K could be the equivalent which is 0 1 & 2 so those are all the possible subsets we could make from set J now if you notice everything is both a subset and a proper subset until we get down to the equivalent set or the evil twin so the equivalent set 0 1 & 2 is a subset of J but it's not a proper subset of J because if it's equivalent it's not a proper subset so the easiest way to think about subsets versus proper subsets is this table the only real issue comes into play when the sets are equivalent if they are equivalent then it's a subset but not a proper subset ok just briefly talk about infinite subsets now some sets have an infinite number of elements it'll be written like this so this would be set I it could be 1 2 3 and then it goes on theoretically into infinity so an infinite set does not have a finite number of elements it keeps going on to infinity let's talk about set notation for certain outcomes we often use set notation to represent a list of outcomes for say an experiment so let's say we flip a single coin well our set of outcomes would look like this they could be heads or tails or a single die roll our set of outcomes could be one two three four five or six for each side of the die if we're looking at a stock's movement from day to day or even hour to hour or minute to minute if you're a day trader that the set of outcomes could be the stock goes up the stock remains even or the stock goes down now look at this one if we flip to indistinguishable coins okay so we don't pay attention to the year on them or something like that we just treat them as if we can't tell them apart so whatever our set of outcomes here well we could have heads heads heads tails or tails tails now why did I highlight the middle one in red well if we're using coins that are indistinguishable heads tails or whatever if there's only one outcome for that so since we can't tell the coins apart it's just heads tails but what if we flip two coins we can tell apart what would look like this we could have heads heads I remember the order doesn't matter so that's one outcome heads heads or we could have tails tails same thing there but look in the middle we could have heads on coin one tails on coin two or heads on coin two and tails on coin one because those are two different coins so we would have this extra outcome so we often use set notation to represent a list of outcomes for an experiment now set builder notation is sort of a way of writing out what we want in our set so for example we have a set C and what we're saying is that it's full of elements and these elements are odd integers less than ten so n is an odd integer less than ten so knowing this description we can come up with the set C that is one three five seven and nine and that's called set-builder notation we don't use it a whole lot but you do see it sometimes in your work or it comes up when you least expect it so this one could be a set P and n is one of the eight planets so set P would be Mercury Venus Earth Mars Jupiter Saturn Uranus and Neptune of course Neptune is my favorite planet so there it is so you can see that set-builder notation is sometimes an easier way to represent sets and sort of plain English okay so let's just go over a few few things and we'll get our conclusion and wrap this up this remember that I set is a simple way of categorizing things maybe objects people planets customers or whatever based on some common criterion we do this all the time and I'll shoot and I'll list a couple here in a minute or it's done to us all the time whether we know it or not so for example usually we hang all of our shirts together in the closet we create a set of shirts now why do we create a set of them well it's because they all have the same shape the same basic function many of them you put on the same way so there's this category or set of things called shirts and then we put them all together as a set we often schedule our college classes according to some characteristic so maybe time program location etc so for almost all undergraduate students you have a set of classes that are general requirements you have a set of classes that are your major requirements and then you might have a set of classes that are electives now as any academic advisor will tell you and a student looking to get a college quickly will tell you is that if there is some way to take one class that will apply to multiple categories you know called double or triple dipping that's a good thing as it saves you time and money so we have these three sets of classes and sometimes they overlap if we can find out where they overlap we can save ourselves time and money but those are different you know examples of sets if you go on amazon.com and look up an item say you look up I don't know we use Harry Potter can let's say you look up you know Harry Potter and the Deathly Hallows part-2 the movie well down below that you'll see a list of items and of course most of them are other Harry Potter movies but some of them may not be it offers similar items below single items that's because Amazon keeps track of what everyone clicks throughout the site so if people tend to click the same type of items they get put in that similar items list that you might be interested in buying Google offers similar apps in its app store so if you download an app to your Android phone I'm sure it works the same way probably with an iPhone but I don't have one it'll say other people installed these similar apps because again he keeps track of what people download and those common interests so this happens these sets are being built behind the scenes all the time now I'll just remember sometimes nothing fits our criteria so you might go into the App Store and you don't find the app you want or you go into the mall and you look for a cellphone case that has kittens on it but there are none so you walk out of the mall with an empty set and that's okay because of all the cell phone cases in the store the empty set that you walk out with which is nothing it is a valid subset okay so that wraps up this video on the basics of sets counting set operations and certain notations so just remember if you're watching this because you are struggling in a class it's just a positive and realize that the fact that you're on YouTube or wherever else trying to improve yourself trying to learn trying to get better that's what really matters okay so just keep that in mind as you continue forward this keep in mind that these videos are meant for people who are new to finite mathematics so if you continue watching these videos I will go into more in-depth topics on the topic of the video if you liked it please give it a thumbs up but on the playlist or share it it does encourage me to keep making them follow me here on YouTube and or on Twitter so you can get my videos when I upload them so most of all I want you to keep having fun enjoy the opportunity you're having to learn and realize that it will seem when you are not it will all pay off in the end so thank you very much for watching I wish you all the best in your studies and/or in your work and I look forward to seeing you again

Channel: Brandon Foltz

Views: 55,477

Rating: 4.9791665 out of 5

Keywords: finite math, what is finite math, finite math sets, sets and counting finite math, finite sets, finite math tutorial, finite mathematics, mathematical notation, math sets, sets math, set, math, finite, statistics 101, mathematics, anova, operations, proper subsets, brandon c foltz, brandon foltz, subsets, set mathematics, venn, set notation, sets, brandon c. foltz, sets mathematics, hccmathhelp, hcc math help, venn diagrams, union, intersection, linear regression, logistic regression

Id: pcedp8eB4UQ

Length: 23min 17sec (1397 seconds)

Published: Tue Jan 08 2013