alright hello and welcome to a lesson on set theory this is going to be lesson 14 14 if you’re in algebra 2 and in algebra 1.5 it’s going to be more like 20 or 21 I’m actually going to do a geometry review or preview in algebra 1.5 that I’m going to hope won’t be as necessary in algebra 2 because all the people in outer two are either also in geometry or had it the previous year whereas only part of the people now if 21.5 I can say that about geometry so we’re going to be looking at a lot of vocabulary and just a little bit of sort of things that you actually would sort of work on or whatever so this is a very vocabulary intensive lesson please pause and go back over these definitions as you have to I’ll start with a little historical background because honestly this is one of the newest things that I will teach you all year this works roughly a hundred years old 100 125 years ago the person who gets credit for creating it actually died in the 1900s he saw a world war one in fact fact world war one was one of the many sort of sad or tragic things about his life in the modern world we would probably have identified the skies having some sort of chronic depression and gotten him some help and some medication that would have probably helped him a lot so he’s probably had an underlying health condition but the last decade or two of his life he had a lot of mental breakdowns for lack of a better way of putting it he actually died in a sanitarium granted it was kind of a pushy rich I need a break style more so than like the insane asylum but still he had some really serious troubles and they were aggravated by public attacks on his work he was struggling to work with problems mathematically that are now known to be impossible and during the tail end of his life practically everything he really had was destroyed by World War one public attacks was work it’s kind of interesting story in my opinion someone that he thought was his friend and mentor someone that had helped him become a mathematician get a doctorate get a job potentially the University of sorts of things once he started working independently that person publicly publicly criticized his work and said that it was didn’t have value or was worthless or was completely wrong those kind of be troubling things parts of his work are now going to be impossible with all that taken together though since his death it’s the last hundred years or seven mathematics they’ve gone back and reworked a lot of mathematics to include his idea of sets and his ideas of infinity because it turned out they were really good ideas it just took a little while to recognize that it wasn’t immediately recognized during this lifetime set is sort of a undefined term in mathematics but we’re going to agree that it means a list grew for collection of objects that are being thought of sort of this one thing and there’s one those individual things are called elements and so by by example we can say that the set a is going to include one two and three we always are going to use capital letters to name our sets and we’re always going to start and stop or in close our sets embraces these are called braces I home one is a set with one is a part of said a four is not and so you can use this symbol to mean is an element of or as part of and so probably not terrible shock that when you put a slash through it means not so one is an element of a four is not an element of a in real life you use sets to refer to collections as a whole when you talk about encyclopedias golf clubs

or pokemon cards or it’s even hitting it hidden in the word place setting which generally refers to some combination of these sorts of things which you think of them all together you’ve already worked with sets a lot in math um you just didn’t necessarily think of it as sets because the solutions of an inequality or an equation or points on a line or you might have actually specifically heard referred to as a set of data points or the possibilities and a probability experiment like heads or tails for flipping a coin or the numbers in a pattern sequence all those things are now thought of in terms of sets there are individual things that in some circumstances are easier to think of as sort of more of the parts of a whole one of the easiest ways thing about sets as Venn diagrams so you’ve been doing Venn diagrams since probably third fourth fifth grade so everything you’ve ever done with Ben diagrams was actually about sets and you’re probably not terribly shocked why’s it called a Venn diagram it actually comes from someone’s name was this last name they are actually a little bit older than set theory 1881 was when that was book was published that would include set diagrams notice that that would have been well within came towards lifetime so he would have probably been familiar with John viens work a little bit more vocabulary and I don’t think you have to write these down Morris once you have heard the phrase well-defined sets well to find if you can decide whether or not a given element is an element of that set and there’s ideas that sets need to be described precisely and accurately or just list everything that’s in there so here’s an example of not well defined set of mr. carpenter students who like his class that’s not a very well-defined set because liking someone’s class sort of means different things to different people and so that that set could be probably better to find that hopefully it’s a set that would have a few of you in it but about one hundred percent sure which because that’s not what I mean the same thing to everyone and the ellipsis a lot of you call that dot dot dot it actually has a name ellipsis pressure English teachers later today by referring to as an ellipsis sometimes we use those when we’re making lists even within sets which you just have to be careful how you use that which we see that in a moment the next thing with notes page is really about roster form so here are the four examples worked out for you so we’re just going to talk about each one the set V of vowels in the English alphabet well it tells you the name the set V V equals it’s a set so we start with braces or enclose it embraces thousand the English alphabet would include the letters AE I owe you some of you might ask what about why well remember the familiar saying sometimes why so in order for this to be well defined we have to leave out why why is sometimes a Val but it’s not always about the set C of primary colors c is equal to the set that includes blue red yellow for paint or pigment that is your primary colors the set a of letters in the English alphabet A equals start the set give me at least a few letters to get me going use the ellipsis give me a few to let me know that you’re stopping you do not have to list out all 26 letters in fact in the next example that’d be impossible set in of counting numbers n equals start the set 1 2 3 4 5 and so on notice that we’re using the ellipsis a very different way here then we are here instead a we’re using the ellipsis to leave out some of the middle that we feel that the reader should be able to fill in here we’re using the ellipsis to show that it just keeps going forever the null set or empty said I actually prefers the phrase empty set I feel like it’s more descriptive although null is a good word that does mean nothing or knowing contains no elements I prefer this symbol for empty set a lot of you have used that in the past and other ways and mathematics

please only use it as empty set seeing people use it for like no solution and that’s not I understand that usage but it’s not one hundred percent correct that is actually the Greek letter Phi the universal set well this might make you think of universe society that came to all possibilities within a situation or context and we’re going to use capital u for university set so we’ll never name a set you unless it’s the universal set that we bad manners to call some set you just because you’re talking about unicorns or something else that might start with you we’re going to agree that we always use you for universal sets finite if it has exactly in things in it where we’re in zero or a number you can count to so the idea of finite is is definite certain specific on the previous example here all of these were fidi except for the last one this one has five elements this one is three this one is 26 this one however is infinite it has infinitely many things in it so that’s our other option is an infinite set the null sets finite because you can count how many things are in it 0 depending on the context the universal set might be finite it might be infinite it just depends on what you’re doing this is kind of a tricky word in mathematics equivalent two sets are equivalent the prefix equ I should make you think of equal but it’s not the exact same is equal two sets are equivalent if they can be placed to go one-to-one correspondence meaning you can pair one thing from this set with exactly one thing from this other set over here they have the same number of elements basically two sets are equal if they have the same elements and they don’t have to be listed in the same order sort of a tricky example of this would be if you did the letters ABCD capitalized like here but then you did another set where it was a b c d and it was lower case technically as far as a mathematician is concerned those are only equivalent because the difference between capital and lowercase might actually mean something so we don’t want to assume they are actually the same this is the part that sort of gives people trouble it’s called set builder notation and it has this basic format set s is equal to the set of all things X you read this vertical line is such that or where and then you put some sort of property or description over here and so the examples I give you on the notes are you know you get e is 2 4 6 8 10 and so on one well most of you can figure out that a good description for the things that said or even natural numbers and so we would write in set builder notation is equal to the set of all elements X such that X is an even natural number a slightly different way of writing the same thing that I do want you to see as you could say oh let’s switch from ballpoint all right you say the set E is the set of all elements X where X is an element of our natural number set um such that you know X is even it’s a slightly different way think I mentioned this back in a lesson I’ll numbers that’s what we talked about the idea that that’s a standard symbol for natural numbers it’s more specifically the set of natural numbers whereas Z is integers that came from a German word for number q was rational from quotient and then real numbers was in it for the set of real numbers it’s a kind of a capital so people call this font blackboard bold because if you want to write in bold on a blackboard you can’t really do that to use these extra lines so some books would have these symbols

just involved the other example was set builder notation was p equals Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune most of you of course read through that you pretty quickly figure out well that’s just sort of a list of the planets we got to be a little more specific so in set builder notation I wrote p equals the set of all X such that X is a planet orbiting the star sold because that’s one of the technical names for our son or X as a planet orbiting our Sun five to ten years ago if I were teaching this I would’ve had been is particular here because at the time we didn’t know of any planets elsewhere we’ve since scientists that’s taking care of that for us though we now know of dozens if not hundreds of what’s called extrasolar I believe planets planets through outside of our solar system and we’ve also in the last five to ten years kicked Pluto out of this list because we’ve also just found all these basic the dirty ice balls out there sort of in the same region as Pluto and so you kind of either have to kick out Pluto or throw in a bunch of things that we really don’t want to throw in as planets all right on these I really want to point out to you the Venn diagrams are going to be extremely helpful and then I also put something up here in symbols that those of you that going to see this sort of more to college level you’ll see this again but the Venn diagram is probably what’s going to help you understand it right now the idea of a subset well sub means like under or beneath so this is like a set under beneath another set well what’s that mean it means that one set is inside of the other so you would say a is a subset of B or you could say if you’re more focused on being you could say that set be contained set a some sort of fun with subsets this is where you start getting into like kind of seeing maybe why can’t oars ideas were a little bit crazy at the time first ones not too terribly shocking any sets of subset of the appropriate universal set fine I get that the empty Saturn all set is a subset of any set in English what you would say is that everything contains nothing everything contain the second bullet is saying everything contains nothing which sounds a little funny at first and he said is the subset of itself so it’s sort of an English or in everyday words you’d say that everything contains itself which can might sound a little funny at first it gets even worse when you start thinking about what happens if you take this is you know known to be the empty set well what happens if you put that inside of a set well this is not empty so please don’t ever oopsies and do that because that’s technically not empty that outermost set has one thing in it my standard analogy for this is you know pretend this is a bag well the bag can be empty pretend this is a try can the trashcan can be empty but as soon as you place the bag in the trash can the bag still empty but what about the trashcan well the trash cans no longer empty it’s got one thing in it and so that’s kind of what’s going on here either these two symbols mean something empty but if you accidentally put one inside the other it’s anymore the bags empty the trash cans empty but a bag inside of a trash can well the bag is still empty but not the trashcan anymore and some students find that kind of humorous and some students think that that’s me just being ridiculous a compliment of a set and by the way disjoint does come up in a minute the complement of a set I’m going to choose to use this pond notation although I’ve

also seen over lining and little superscripts of see it’s one of the few things where there are three or four different symbols running around and they never come to an agreement about which one should be used the complement of a set includes everything else and so if this is some set and this is the universal set then all the stuff over in here is the complement kind of like what’s the it’s kind of like opposite of so speak if you’re looking at all the students in the class what’s the opposite of the boys two girls and so the idea is that the complement of a set is set of all X that are possible but not already included in the set and so not too shockingly what’s the opposite or nothing everything the complement of the null set is the universal set what’s the opposite of everything nothing so the complement of the universal set is all set oh by the way please don’t just try to write down the definition also do either your own version of this bond Venn diagram or copy it because I tell you I promise you the Venn diagrams are probably what most of you are going to think of when you’re trying to do answer questions about this we’re going to set difference it’s supposed to make you think sort of like subtraction you take set a you cut off the part of set B you cut off a part of set a that’s also in set B and throw it away and what you’re left with is sort of like pac-man shape right here is a set difference and so you say that a minus B is set that includes any element of set a but is not also an element of set B and so in set builder notation it looks like this a minus B is instead of all animals X such that X is an a but not in B and sort of my open question I’ve asked you here is how does the Venn diagram change if we do a different subtraction B minus a I think most of you can pre quickly tell me what would happen there if not think about it intersection might make you think of like a road where two roads cross or overlap and so it’s kind of more like what people are used to within diagrams is here’s two things shade the intersection it includes any element that’s a set of both set a and set be any element it’s a particle setting Part B so our symbol for intersections can become this upside-down you or tell some people to think of it it’s like the kind of like the end but not exactly in as an intersection kind of like intersection here and so it would be set of all elements X such that X is an element of a and X is an element of big excuse me kids stuff in the way so disjoint sets have no elements in common so my question for you here would be what’s a Venn diagram look like for disjoint sets how can we draw is your set a and stupid EV do you think about it for a second and maybe you realize that you could sort of make it look like a brick this is disjoint sets disjoint join us in together this is a not not together sets that have no elements in common they don’t overlap at all it will Union well the Union kind of tells you why we use that upside down you prefer i use you for union includes any element in from either set a or set B you join the two sets together so a union B is the set of all elements X such that X is an element of a now the key words or X isn’t wanna be so for intersection we’re looking at the word and and the overlap for union we’re looking at the word or and it’s all together I got everybody else in bed I’ve got to finish this example for you off first thing at least when you’re first getting used to these sorts of problems where you’re actually trying to use this stuff sorel is that we’re just looking at you know a list here with some numbers and so we might go ahead and make a Venn diagram just to kind of help us out and try to see if I can i get 1 sketched right in sort of this area we’re going to use the box to be the universal set and so i’m going to use a circle or an oval or just any sort of amorphous blob it doesn’t really

matter what it looks like to represent set a here so and it’s going to of 235 so they’re setting be needs to have two and four and the twos already right here that’s B and let’s say two three four five the only things left out of the list that’s an option is one and so one can just sit or be floating over here by itself just hanging out and then we’re going to use that picture to help us work out each of these problems so when you look at this Venn diagram or you look at these sets what are the things not an a what’s a compliment and hopefully you can convince yourself with the things that are not in a r1 and for the set containing one and four B complement 35 and one and we agree to generally write things in order not that it matters okay so which one is the so this is intersect and if you look at picture the only thing they have in common the overlap is just too then this one’s Union because it looks like the EU and again that’s kind of the idea a union is two things coming together and becoming like one thing like marriage is a legal legal union or during the Civil War the Union referred to the states that wanted to preserve the United States and keep it all together and just like when you get married if you if you have two toasters it doesn’t really matter you just get rid of a toaster you say well we have a toaster and so we’re going to take sets a and B we’re going to put them together and it doesn’t matter that they have a 2 in common all we need is one copy of everything they have well everything they have would be 42 or 43 of 52 2 3 4 5 if you look at sort of this region of the Venn diagrams imagine that becoming one big circle it would have two three four five a minus fees you look at a and then you’re going to cut off the two and throw it away so a minus B is 35 b minus a on the other hand if you do the subtraction in the other order where you start with be four and two if you cut too often throw it away so B minus a is just for and have a worksheet that will probably end up doing in class with in your groups where you just got a bunch of these problems and they’re mostly either true or false or just fairly straightforward try to come up flowing a little answer type questions so that we get some extra practice with this since this is completely new to almost everybody but don’t underestimate the importance of just getting down like what these symbols me if you can read the notation and understand what it means you’ll find that it’s fairly straightforward and easy for most of the things that I’ll actually ask you today thank you and see you next time