Set Operations

– [Instructor] Okay so this is 2.3 set operation This is our second video So we have some new objectives Our objective is to find the intersection of two sets So we need to find the union of two sets We need to find union and then perform operations with sets We are gonna find the intersection and the union of some different sets We determine sets involving set operations from a Venn diagram We draw a picture involving the intersection and union of two sets and determine which elements are a part of that set We are gonna understand the meaning of ‘and’ which is intersections and ‘or’ which is gonna be union here and then we are gonna use the formula for finding the cardinal number of the union of two sets and we are gonna use a Venn diagram to help us understand that On to the definitions, so the intersection of sets we are gonna have our definition The intersection of sets A and B written, A intersect B is the set of elements common to both set A and set B So because it has a common element in both as part of our Venn diagram, the idea is the intersection is this second region in-between the two All right, so that’s the overlap right there So generally this is gonna be a smaller set because it has got to be in both It has got to have characteristics of both This definition can be expressed in the set builder notation but I think this notation is very nice right here So let’s do some examples of finding the intersections of two sets Now if we are gonna find the intersection here, what we need to do is we need to find elements that are in both Seven only shows up here, but I do have eight in this set and in this set So I’m gonna put my brace in and put the eight in All right and then eight I have in this set, but not in this one 10 I have in both sets, so that’s gonna be in the intersection of course and then I have a 11 over here, but not over here All right you’ll see the six and 12 do not show up in this other set either The intersection of my two sets is simply eight and 10 And you’ll notice that it is smaller than the other two sets, all right, either one of them is a cardinal number What about this one? Okay so one is here but not over here Three is here, but not over here Five is here but not over here Seven is here but not over here Nine is here but not over here If you look, these are all odd numbers and these are all even numbers so there is no overlap and so if this is going to be the empty set, there are no elements in either one and that’s certainly smaller All right, now here is just a nice property Any set A intersected with the empty set will be the empty set There are no elements in the empty set, so we can share none of these elements with any other set So this is always gonna be the case so this is the answer to C right here and it is a kind of property and that makes us happy What about the union of sets? So the definition of the union of sets is the union of sets A and B written A union B see how it looks like a U is the set of elements that are members of set A or of set B okay and so this definition can be expressed in set builder notation It is also pretty nice to see the Venn diagram right? Here of course is set A Here of course is set B And of course all of these elements, let’s change color All of the elements in A, or all of the elements in B are part of it You are like oh so it is all of this All of this area All of these elements are part of this new bigger set So unions are almost always bigger sets instead of smaller sets And that’s because of the or as opposed to when we had the intersection of our sets

that was the “and.” That was one of our learning objectives right so we want to make sure that we understand that There is our union of sets And now let’s do some examples So here we go Find each of the following unions Remember it is every element here and every element here, but do not repeat elements This is a big mistake that students make is they repeat elements I’m sorry that got squished That makes me sad I’m gonna put my braces in for the start of my set and I’m gonna put well six is the smallest one but we don’t have to go in rank order So seven, eight, the eight is already over here so I’m not gonna write it again Nine, 10, 10 is already over here so I’m not gonna write it again, 11, 12, and then the six right? Now if I really wanted to, I could raise just a little bit here and we could go back and we could put it in order and that would make me happy but remember they don’t have to be in order The elements don’t have to be in order But there they are Now if we look at this union remember these are odd set, again this is our odd set This is our even set again Now we are gonna union them, so we are gonna put them both together and so this one is just gonna be one, two, three, four, five, six, seven, eight, nine okay? That’s our new set right there That also makes us very happy Now we have a new property Now we have the union with the empty set right here and now with the union and the empty set it is kind of like adding zero Doesn’t change anything So a set A union with the empty set is set A It is not gonna change any value So here the answer is one, three, five, seven, and nine That makes us real happy because it is like our zero property of addition So that’s great So we are just gonna summarize that now We’ve already talked about it but any set A intersected with the empty set is the empty set Of course the A union with the empty set is just A So this is kind of like multiplication If this is the analogy for it because remember A times zero is always zero and this is kind of like our addition Here because A plus zero is just A So you see a very similar idea in the addition and multiplication with the intersection and the union This is a little bull’s eye connecting things to what we already know So now performing some examples on the sets So what are we gonna do here So U is one through 10 That’s our universal set So now we can talk about the complement here right? A is gonna be our one, three, seven, nine B is gonna be three, seven, eight and 10 and then we are gonna remember to always perform any operations inside parenthesis first Okay this is our orders of operation and of course parenthesis first our grouping symbol first is super important Okay so here we want to union A and B and then take its complement So let’s first find A union B and then we will take its complement So A union B is gonna be one three, the three shows up in both but I only need to write it once Seven, the seven shows up in both so I only need to write it once Eight so I go in order

Nine, and then here is 10 So that’s A union B So I put those two sets together Now I need to take the complement So A union B’s complement is everything not in this set but still in the universal set As I look, one is in the universal set two is in the universal set but not in A union B, three is there, four, five and six are not so we want to write four, five, and six And then seven, eight, nine, and 10, seven, eight, nine, and 10 are all in A union B so those guys are not going to be in the complement So there is the complement and that makes us happy I find it very helpful and hopefully you will too I find it helpful to cross things, elements, out as I go Now this is a mess So what I’m gonna do now is let’s go ahead and erase all of this ugliness right here and then in a different color we will do on this one Now you want the complement of A intersected with the complement of B Really what I need to find is A complement first so everything that’s not in A that’s in the universal set As I look, one is in A, so I don’t want to use that Two is in the universal set, but not in A three is in A, four, five, and six are not in A, four, five, and six are not in A but they are in the universal set Seven and nine are not in A so eight and 10 right? Eight and 10, great Now we need B complement so that then I can do the intersection So B complement will one and two are in U but not in B One and two are in U but not in B Let’s see between three and seven so four, five, and six again Four, five, and six again Then you have seven and eight and then nine is missing So we need nine from over here So there is the nine So we want to do the complements first and then do the intersection of the sets So the order of operation here is gonna be the complements first and then the intersection So now we are ready to put this altogether A complement intersect B complement and now of course we want to put just those things that are the same together So two is in both, that’s the same in both Two is gonna work Four, five, six, and that’s it Everything else is different That would be the intersection and that makes us happy Not blue but happy Okay so those are our operations with sets and even some sort of advanced operations Okay so now we are gonna be determining sets from a Venn diagram It is very helpful to be able to look at the picture A includes these four rational imaginary numbers B includes these five numbers right here and then of course U contains this additional guy which we will put in red for obvious reasons Out here this part of the universal set but not A and B So now we want to be able to do is to look the Venn diagram and determine what A union B is

So we are gonna set up our braces here and A union B is going to be all seven of these numbers and the only number we are gonna exclude is 666 because it is outside of A and B So we are gonna have Pi We are gonna have E We are gonna have the square root of two and we are going to have the square root of negative one our imaginary number Then we are gonna have in B we are gonna have E to the Pi I which is called Euler’s number We are gonna have 10 to the 100 power which is not quite a googleplex but pretty big number And then two to the Lth, so we’ve got our special Lth naught here which is our natural number right and then we want to close up our set So this is our seven values for A union B Now A union B complement is A union B and then everything outside of it but still in the universal set and the only thing that is there is like we said before the 666 We will put that guy out there because that’s a special number too These are all special numbers so that’s a special number I guess So okay so lets go ahead and look at C and D right and do these two examples right here and so you have A complement is everything outside of A So A complement is going to be all of these so there is my A complement, everything outside of A, intersected with just B Just B is these guys right here And so you are just looking at the set of elements, here is constant, E to the Pi I, 10 to the 100 power and two to the Lth naught So that’s it Then of course A union B complement is slightly different because remember the union makes everything bigger so let’s raise our notation here and take a look at this A is all of this so we are gonna have all of that and then union with B complement Everything that’s outside of B okay so you want to have everything that’s outside of B What’s outside of B is over here and these guys right here And then of course what’s inside A are these two guys right here Okay so we are gonna have these five elements altogether So you are gonna have Pi and E and the square root two, and the square root of negative one and then you are gonna have the 666 The only thing that we are gonna exclude are going to be these three guys right here because they are outside of A, they are outside of A and they are inside of B so they are not part of B complement so those three guys right there are excluded Okay so again circle make notations on the graph I think these are all things that are really helpful with the Venn diagram to help us see what’s inside, what’s outside and those kind of things Don’t be afraid to draw the picture and make the circles okay Now sets and precise use of everyday language Remember set operations and Venn diagram provide precise ways of organizing, classifying, and describing the vast array of sets and subsets we encounter every day So this will be part of your project And as we’ve said before, the ‘or’ refers to that union of the sets and the ‘and’ refers to the intersection of the sets You see these two words you can translate them into these two mathematical symbols when we are talking about sets Finally, the cardinal number of the union of two finite sets Now why is there a special formula for this You are gonna find the cardinal number A union B Well lot of times A is gonna look like this We are gonna use a Venn diagram And B maybe has some elements in A We are gonna go like this

and this is our B So what happens is this is gonna be equal to the cardinal number of A All of the number of elements in A Here’s all of my elements in A And then we recall that N Then we are gonna look at ‘and’ we are gonna add all the elements in B All the elements in B are over here These are all of the elements in B We will say that’s like M You are looking at, well, well, well there’s some stuff in here that we’ve counted twice I’ve counted some of these guys twice in this intersection in the overlap In the intersection of those two sets is some overlap that we need to subtract off We need to subtract off the overlap from the intersection and just one of them So basically what happens here is if I subtract out this oops, if I subtract this out, put the B back What I’m gonna do is I’m just gonna subtract out just maybe the A parts that I’ve counted twice and that will leave me just the B parts and now there is no overlap and that makes me happy Because of overlap, I’m gonna count elements twice and so I need to subtract off once So I’m gonna subtract once And that makes me happy So that’s the idea Now if there is no overlap then this is zero You don’t have to subtract anything If B is completely inside A, if we had something crazy like here’s A And then here’s B then what we have is you end up subtracting all of B out You don’t add anything to it And that’s kind of weird That makes those deep (mumbling) All right so there we go So now let’s work an example Let’s actually instead of just looking at this theoretically let’s work an example Here is our example Example A, cardinal number of the union of two finite sets Now this is gonna be very much like your survey right? So again we did an example like this in class but let’s take another look Some of the results of the campus blood drive survey indicate that 490 students, 490 students were willing to donate blood 340 students were willing to help serve a free breakfast to the blood donors and 120 students were willing to do both How many students were willing to donate blood or serve breakfast? Well let’s take a look right? So what do we know? We know that there were 490 blood donors We know that there were 340 breakfast servers We know that 120 that were willing to do both That means that these particular individuals are in this category and this category This is our overlap right here so maybe it is helpful to actually draw a Venn diagram What if I actually drew a diagram of blood donors and I drew a diagram of the breakfast servers and in the overlap of the 120 So over here, are all of the people who are willing to serve breakfast including the 120 That’s going to be your 340 that were willing to serve breakfast minus your 120 that were willing to do both and so you are gonna end up with 220 people that were willing to just serve breakfast including the 120 people that were willing to serve breakfast and donate blood That ends up with your 340 over the breakfast That’s where that number comes from

I can do the same thing with the blood donors I’ve got 490 blood donors A 120 of them were also willing to serve breakfast and so you end up with 370 that are blood donors only and a 120 that are blood donors and serving breakfast So the actual question was, how many are willing to donate blood or serve breakfast? That’s each of these three added together Because I’ve already taken out the overlap These 370 are only in here, 370 are only in here The 120 are only in here The 220 are only in here So I add up those seven, eight, nine, 10, 11 carry the one, four, five, six, seven, 710 So that number makes us happy What about our formula? Well let’s take a look What if you had taken the 490, all right, what if you had taken the 490, that’s the cardinal number of set A, and added the 340 together so there’s your zero, nine, 13, carry the one, 830 and then subtract the overlap We just want to subtract the overlap so subtract 120 And you get 710 It is as if math works forever in all ways You can draw a picture like we did here just being very careful about where each thing goes or you can just use the formula of adding up sets A and B and subtracting off the intersection that’s in both and you’ll get the same number either way All right that concludes the video Go on to 2.4