Finite Math: Venn Diagram Basics

you hello and welcome to the next video in my series on basic finite math now a few things before we get started number one if you're watching this video because you are struggling in a class right now I want you to stay positive and keep your head up if you're watching this it means you have come pretty far in your schooling up at this point you're smart and you may have just hit a temporary rough patch now I know that the right amount of hard work practice and patience you can get through it I have faith in you many others around you have faith in you so so should you number two please feel free to follow me here on YouTube and or on Twitter that way when it upload a new video you know about it and on the topic of the video if you liked it please give it a thumbs up share it with classmates or colleagues or put it on our playlist because that does encourage me to keep making them on the flipside if you think there's something I can do better please leave a constructive comment below the video and I will try to take those ideas into account when I make new ones and finally just keep in mind that these videos are meant for people who are relatively new or brand-new to finite math so I will just be going over the very basic concepts and I will be doing so in a 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 the next in our series on sets and counting usually this is one of the first things you do in the first few weeks of a finite math class now some universities do it differently but where I was at Indiana University this was one of the first things that was done in the first you know two three weeks so this video is about Venn diagrams a lot of students freak out when they start working with Venn diagrams even students who had AP Calculus in high school because it requires you to think in a different way but the reality is we do this all the time in real life whether we realize it or not and maybe in this video and definitely in the next video I'm going to talk about how we actually create sets and subsets and intersections and unions and everything in our everyday lives without thinking about thinking that we're actually doing it so let's go ahead and look at some problems just a quick review from our previous video about what sets and elements are so remember we have a set and we're going to call this set in now you can think of the outside is kind of like a fence so everything inside of it is part of set in and in this case it's the numbers 1 through 9 so 1 2 3 4 5 6 7 8 9 are sort of inside are set in now we use the is an element of sign to denote things that are inside our set for example the number 4 is inside our set it is an element of n an element of set in however we also have the opposite you can see that 12 up there in the right hand corner 12 is not a part of set in it is outside of its boundary therefore we say 12 is not an element of M so it's the element symbol with a sort of slash through it so those are just some very simple reviews of what sets our what we call the things in them which are elements and how we denote things that are not part of a set now just a quick review about equivalent sets so this set is actually two on top of each other so I've kind of used a checkerboard pattern to try to show you that there are two sets here but they are really sort of right on top of each other now the thing is they have the same elements so set n is 1 2 3 4 5 6 7 8 9 and so is set m so N equals M and M equals n a Evelynn sets now the previous video we talked about subsets and proper subsets now in this case equivalent sets are only subsets of each other okay they cannot be proper subsets so if you're not sure about that difference please take a look at the previous video we did but these are subsets of each other so subsets and proper subsets so here we have set in and inside of it we have nested this other set II now just by looking at the picture you should be able to tell that II all of it the whole thing is inside in so think about this whatever is in N is everything in E in the gray part and everything outside of E in the orange part that's in so we say n is 1 2 3 4 5 6 7 8 9 but what about e well e is just the numbers 2 4 6 and 8 so E is a subset of n so we can write it like this e is also a proper subset of n which we write like this now just a quick review as to why that is see it can be a proper subset because it's not equivalent that's sort of the thing that you know forbids equivalent sets from being proper subsets so E is both a subset of N and a proper subset of n again if you need to review those just take a look at the previous video so I'll talk about intersection that's where these problems become interesting and this is probably how your problems are set up in your class so here we have N and here we have E so set N and set E now if you notice they overlap here in the middle but the thing is is that neither set is totally inside or nested inside the area of another so there is no subset here there will be some elements of em that will be outside of e so if you look at the diagram here everything that is in the pure orange color is only an M now there will be things that are in E but outside of N so the part of the diagram that's just blue those things are just in E but what about the middle part sort of the purple where the colors come together well that is our intersection and anything in that area belongs to both it belongs to both sets so let's take a look at that further let's say we have a set a and it's the numbers 1 through 9 then we have the set B which is the numbers 6 through 14 so 1 through 9 and then 6 through 14 so the question is what elements do these sets have in common what elements do they share where do these two sets overlap well if we look they have 6 7 8 and 9 in common those are the elements they share now if I write these sets so they actually overlap so I line up the sixes I line up the seven line up the 8 line up the nine what I can do is actually show where they overlap where they have in common so I can put a little graph there so 6 7 8 and 9 or where they have their elements in common where they overlap exactly so 6 7 8 and 9 belong to both sets the numbers 1 2 3 4 and 5 only belong a so only in a and the numbers 10 11 12 13 14 are only in B so we have sort of three regions here in the middle we have the intersection we have the overlap the things in common which are 6 7 8 and 9 then we have numbers that are only in a 1 2 3 4 5 and numbers that are only in B 10 11 12 13 and 14 now if we actually put this in the Venn diagram it looks like this so as you can see in the middle we have 6 7 8 & 9 the area that's only a sort of the very dark orange part we have 1 2 3 4 & 5 just like our graph above it and then over on the right we have the numbers 10 11 12 13 and 14 just in the pure blue over there on the right now we do have to make sure that our original sets are intact just make sure we look at set a so do we have the numbers 1 through 9 will have 1 2 3 4 5 in the just a part and then we have 6 7 8 & 9 in the overlap part so if we kind of put heart our hand over the just blue part where B is we can see that a does indeed have the numbers 1 through 9 now check B does it have the numbers 6 through 14 well it has the numbers 10 11 12 13 14 in the just B part in the blue and it has 6 7 8 9 in the intersection so if we put heart hand over the orange part and we just look at the intersection and the pure B part we have the numbers 6 7 8 9 10 11 12 13 14 so again we can look at it a couple different ways to show how this intersection is represented in just a general sort of graph up there in the upper right or in the actual Venn diagram in the lower right so a little bit more about intersections what we say is these are the elements these two sets have in common now another way we can say is that these elements must be in a and in B in a and in B so a is one two three four five six seven eight nine B is six seven eight nine 10 11 12 13 14 so what elements are in both what do they have in common so we say a intersect B and remember that's the intersect symbol down there so we're upside down you a intersect B equals the set 6 7 8 9 or that intersection and that's how we write that so this is how we do very simple intersection using two sets that overlap and have elements in common now what about unions so intersection is what they have in common Union is a little bit different now the way we put this is that the elements in just a there in the orange just be there in the pure blue or both and that's the intersection in the middle so anything that's in just a just B or both so we can also say the elements must be in a or B now in mathematics we use the or as inclusive so we really are saying they have to be an a or in B or both so a is 1 2 3 4 5 6 7 8 9 so this is this region right here the a circle outlined in the green B is this area now notice that down in the be set down there I crossed out the 6 7 8 & 9 well why did I do that because when we're doing a union we're interested in sort of what's in the whole thing but we can't count those numbers twice we're only going to count them once so when we say what is a union B it's all the numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 and 14 so all those numbers appear either in just a just B or both or the elements are in a or B or both and we don't double count the intersection where they overlap now in the next video where I talk about cardinality and that's where this comes into play when we can figure out what parts of our diagram we are missing just keep in mind that the union is basically anything that appears in that region once then we just count that up and that is our a union B so if we really kind of take everything out but numbers and the color is nothing like that the union is basically this shape here so depending on where you are in the world you may or may not have what are called peanuts but we have those here in the US as this is kind of a peanut shape or some people call it like a bean shape so we have what's an a what's in B or what's in both if we kind of again erase all the lines this is the sort of overall area we're interested in now disjoint sets there is no subset okay there's no intersection they're completely separate so in this case in set in intersect E okay is no it's empty so there is no intersection between N and E and of course there's no subset because they don't overlap in any way so these two sets have no elements in common because they do not overlap and there are no subsets or anything because one is not inside the other but remember an empty set is a valid set it's actually a very important set in many instances but an empty set is a valid set so you might get a question that says what and here's the diagram and they go well what is in intersect e and the answer isn't zero the answer is not does not exist the answer is not whatever the answer is empty and intersect E is empty it's an empty set now let's talk about the universal set so basically the universal set contains all the elements were interested in and it really just depends on the problem so you might have a problem that's about all the clothes and a store or all the students in an academic program or all the restaurants or other businesses in a certain region in the u.s. so the universal set kind of puts a boundary around all the things that will be under discussion or that we are interested in and again it really just depends on the problem now inside that Universal set we can have all kinds of subsets and intersections and unions and all kinds of stuff so think about this everything that is in s some of them are also in a here so all of a is inside our Universal set they'll probably be some things in a and some things outside of a so if an element is in s but it is not an a so it's sort of in the orange floating around outside there then we call that the complement of a which we didn't note with a prime and we'll put numbers in here in a second so everything in s can either be an A or not in a now there will be nothing outside of the universal set so everything in the universal set has to be an either a or not in a let's look at this look at our Universal set look at all the numbers in that rectangle including what's an a because of course a is in s so we have to count it so that's the numbers 1 2 3 4 5 6 7 & 8 now what is in a well a-alone are the numbers 2 4 6 & 8 of course those also belong to s because it is inside s the question is what is a complement what is in the universal set s but not in a so a prime well those are the numbers 1 3 5 & 7 so that's just again that's just the idea of the complement of a set so whatever is in a is in there of course but there are still things outside of it in the universal set so they are not in a now of course you'll have problems and we'll do them in future videos that involve the intersection of three sets so this is a very common graphic you're going to be looking at doing all these Venn diagram problems now I just kind of started to give you a hint of some of the things you're going to be finding so if you look in the middle here we have set a B and C and right in the middle is where they all three overlap so a intersect B intersect C now here are a couple bonus questions what regions are a intersect B so going to be this careful this you have to develop a vision for doing this so a intersect B let's look at a the circle for a then we look at the circle for B now what region and what regions do they intersect well one region they intersect what we have labeled a intersect B intersect C there's also one more region where they intersect you see it okay that is the region to the upward left of a union B Union C so it's kind of a darker green color in the upper left that's also part of a intersect B so basically what this becomes is like a puzzle you have to solve then diagrams where we have missing values and we're given certain values we have to use set relations and set operations to sort of figure out the missing pieces of a puzzle I've always wondered why puzzle books don't have Venn diagrams in them because that's exactly what they are it's all about finding the missing pieces so all the parts add up to what we know they should be what we have in the problem etc but we'll do these in future videos okay so that's a very brief straight-to-the-point introduction on Venn diagrams so again we learned about intersections we learned about unions learned about disjoint sets where we viewed sets and subsets and things of that nature so this is just meant to get you a firm foundation to do more complicated problems involving Venn diagrams and trust me they do get very complicated they get very I don't know logically where we have all kinds of sets and parentheses and complements and crazy so you'll see how they work but the thing here is you have to develop a vision for looking at Venn diagrams you have to think of them in pieces sort of almost in layers and sometimes when the students I work with actually had to write out the pieces or separate out all the regions of the Venn diagram and then sort of put the pieces back together you know one by one that's what it takes that's what it takes that's completely fun so again it's just about developing a vision in front of sense of how they work okay just a few reminders if you're watching this because you were struggling in a class I want you to stay positive and keep your chin up you are very smart talented person some of this may be new to you and it may just be temporarily you know sort of challenging you which is good that's a good thing now I know what the rhythm of hard work patience and practice you can get through it again you should have faith in yourself because me and other people around you at school and at home and everywhere else do have faith in you too so you can do it please feel to following me here on YouTube and on Twitter so upload a video you know about it if you liked the video please give it a thumbs up most important of all just keep in mind that you are dedicated to beyond here learning trying to improve and that's what matters so thank you very much for watching I wish you the best of luck in your studies or in your business or whatever and I look forward to seeing you again next time you

Channel: Brandon Foltz

Views: 51,852

Rating: 4.8850298 out of 5

Keywords: venn diagram basics, venn diagram tutorial, finite math, universal set, finite mathematics, venn diagrams, union, intersection, brandon foltz, brandon c foltz, brandon c. foltz, venn diagram, venn, sets, math, diagram, statistics 101, anova, disjoint, finite, basics, video, set operations, probability, partition, complement, equally likely outcomes, tree diagram, set, subset, hcc math help, logistic regression, mathematics, statistics, hccmathhelp, linear regression

Id: V1J-4YFK7FQ

Length: 23min 22sec (1402 seconds)

Published: Thu Jan 10 2013