Hey, hey! Venn diagrams time! Today we're going to take what we know about sets and we're going to start drawing some pictures.
So this is where it gets more fun. So we're going to be able to use diagrams to help us to visualize what's happening with our sets. Okay, what is a Venn diagram? Many of you have probably already seen one before.
Maybe not in math class, but possibly in other classes like English or science. It's a useful visual tool for showing set relationships. Consists of a rectangle that holds all the members of the universal set, and then circles within the rectangle to hold the members of various subsets of the universal set. So in the rectangle, we've got the whole set, but the set can be subdivided into the circles. This diagram is named for John Venn, an English mathematician and their inventor.
There are lots of different ways to draw Venn diagrams. We're going to be focusing on ones that have two circles in this lesson and then we'll look at three circles in the next lesson. You can always draw a Venn diagram with the circles overlapping and just fill it in from there, but you don't have to. You have some flexibility depending on what's going on.
I mean you could always draw your Venn diagram like this. Universal set and then like subsets A and B you can put in the circles here. looks like the MasterCard symbol so that's all of these I'm going to show you you can always draw them that way but I just want to show you that there's other ways too okay let's say you have sets a and B where set a has elements like one three five and set B has elements two four six When you go to draw the Venn diagram, you could draw it with the two overlapping circles. But notice that, you know, sets A and B don't have anything in common.
So there won't be anything inside that overlap. Set A, the circle A will have 1, 3, and 5 in it, but 1, 3, and 5 will not go in that overlap because they're not in common with set B. And then set B has two... four six and that will not go in the overlap either because set a doesn't share those in common so that is one way to draw it but there's another way to draw it for these disjoint sets sets they have no elements in common.
You could draw the circles completely separated and that way at a glance people can see oh these don't have anything in common. So that's perfectly acceptable way to draw it. So they don't have to overlap in the case of disjoint sets if you don't want them to.
subsets let's say you have set C that contains 10 11 12 13 14 and let's say you have set D that contains I don't know why it does that line two things that contains 10 12 14 Well, one way to draw the Venn diagram is to have the overlapping circles, the MasterCard symbol. And in this case, the 10, 12, and 14 would go in the overlap. because that's what C and D have in common.
Then C additionally has 11 and 13. That's not in common with D so that'll go over here. D does not have any extra elements besides the 10, 12, and 14 so that would be it. But when you have scenarios like that where essentially D is a subset of C, because all of D's elements are found in C, or you could say too that D is a proper subset of C, all of D's elements are found in C and D and C aren't equal. Another way to do that is to use nested circles, and that's perfectly fine as well.
So there's a little flexibility in how you draw these. Since D is the subset of C, You make C the outer circle, make D the inner circle. It's kind of like an egg with a giant yolk. So circle D will have the 10, 12, and 14. And since circle D is located inside circle C, we don't need to repeat the 10, 12, and 14. We just need to add the 11 and 13 out here, still in C's circle, but not also in D's circle.
Equal sets, you can always share the same circle. So let's say you have set F that has ABC and let's say you have set G that also has ABC. These are equal sets.
They have exactly the same elements. If you use the MasterCard version, By the way, that's not a promotion for MasterCard. I'm not getting paid anything. I just feel like it's a good way to explain it.
You can use the overlapping circles. Gosh, I don't know why it's doing this crazy stuff. And then the A, B, and C would go here in the middle. Or, if you want... You can just share the same circle since they are equal sets.
Since F equals G, you can just draw one circle and label it with both F G on it to let people know hey this is representing both of those sets and then put the A B and C inside that one circle. So that's an option. In general though, you can always use the overlapping circles. A little more generic example, let's say you have a universal set that contains 1, 2, 3, 4, 5, 6, 7, and 8. And let's say you have a set C.
Contains 1, 3, 5, 6, 7. Let's say you have a set D. That contains 2, 4, 6, and 7. Well, these sets do not fit any of the examples I just showed you as far as they're not just joint because they do have common elements. They're not equal. They're not exactly the same sets. One is not a subset of the other as far as C and D are concerned.
C and D are. subsets of the universal set but that's to be expected remember the universal set is the entire set that we're talking about and then you can have subsets that that you're putting into the circles in the diagram okay so when do the circles for C and D. I apologize for not actually being circles but it's the best I can do. You first figure out what goes in the overlap. So in this case we got six and seven in common.
Then set C also has a 1, 3, and 5 that are not in common with D. So they go out here in this part of C's circle. And then D also has a 2 and a 4 that are not in common with C. The 6 and 7 are already covered in the overlap. Then the universal set, let's see, we've got the 1, the 2, the 3, the 4, the 5, the 6, and the 7. But notice that 8 was not included inside the circles, but it's still part of the universal set.
So it still has to go out here. Still in the rectangle, but not in the circles. That's the part that people forget the most often. They forget to include the rest of the elements from the universal set.
So please don't forget to do that. Alright, so that's some different ways that you can draw the Venn diagrams with two circles. Now we can use our Venn diagrams to carry out various set operations. Set operations include things called complement, intersection, union, difference, and Cartesian product or cross product.
Let's start with complement. The complement of a set, A, denoted A without little mark, so I just read A complement, is the set of all elements that are in the universal set but not in A. On the Venn diagram, the complement is the set of elements found outside of A's circle but still in the rectangle.
So let's say you have a universal set. contains A, B, C, D, E, F. And let's say from that you have a subset A that contains B, D, and F. Okay, when you go to do your Venn diagram, This one's only going to have one circle because you only got the one subset of the universal set, circle for A, that's going to contain B, D, and F.
And then the universal set still has A, C, and E in it that are not part of set A. So in this case, you know set A contains B, D, and F, a complement. It's everything outside of A's circle. So that includes A, C, and D. An A complement is a set.
You'll notice that as we're doing these operations, the answers are sets too. So you've got to make sure you use those little set braces to say that. Kind of a common problem in my math lab people run into. They forget the set braces when they're needed. So just watch out for that.
and then another small example let's say you have universal set containing A, B, C, D, E, F let's say you have a set M that contains A, C, D So you have a set N that contains B, C, E. Okay, when you go to do the Venn diagram for this, you have two circles because there's two subsets of the universal set. M and N. Now M and N both have C in common. Set M also has elements A and D.
Set N has elements B and E. The universal set, let's see, we have A, B, C, D, E. The universal set still includes F. Alright, now with the complement.
Let's say you want to do M complement. The complement is everything else in the universal set that's not in M. So M has A, C, and D, so the complement has to be B, E, and F. You can get that either by looking at your universal set and saying, well, I've got to leave out A, C, D. That leaves me B, E, F. Or you can come over here and look at your Venn diagram. Cover up set M.
Just cover up M's circle entirely, just that circle, and see what's left. And you'll see that outside of that circle, you have B, E, and F. And notice that B and E are in another circle, set N, and F's outside of the circles. And that's okay. The complement is just strictly telling us what is not.
in set M what's outside of that in this rectangle in complement same idea cover up in circle so I'll leave out BC and E that leaves us with a and notice that M complement and complement have F in common because that was the one element outside of the circles Alright, intersection. Intersection of two sets A and B is denoted A intersect B. To me, that little intersection symbol kind of looks like a big fat N.
So to me, I think of it as N for intersection. The set of elements that they have in common, which we've been addressing that in the previous examples, everything that was going in the overlap that made up the intersection. So these elements are found in both sets A and B at the same time. They're in that overlap.
On the Venn diagram, the elements of the intersection are found where the circles for A and B overlap. All right. set haha got ahead of myself alright so universal set let's see here I'm so bad with being creative I keep using the same stuff over and over again let's say that the universal set contains P Q R S T B. Okay, let's say we have a set G.
So your sets can have any letter you want to call them. Let's say you have set G with P, Q, S, T. Let's say we have a set H with R, S. T, V. Okay, so we draw the Venn diagram. I have circles for sets G and H.
Okay, what goes in the overlap? So G and H have S in common and T in common. So that needs to go in the overlap.
Set G also had P and Q, but those are not in common with set H. Set H has R and V, but that's not in common with set G. The universal set is still missing U, P, Q, R, S, T, U, V, and that's it.
Alright, so in this case, the intersection of G and H It's going to be the set containing those elements in the overlap there. The elements they both have, they both have. S and T are found in both G and H.
So that's the intersection. and then union is the other one that's related union you're gonna include everything from the two sets you know I think of Union I think of getting married and my husband and I got married he was already in Jacksonville and I was moving from Lakeland. He already had his stuff in the apartment.
My stuff from Lakeland apartment got moved into his Jacksonville apartment. It all got combined. So, in our union, his stuff, my stuff became our stuff. Or as he likes to joke, became my stuff. Anyways, so yeah, we had overlaps.
You know, we had some things that were the same, so we had to weed it out. Whatever was the intersection, what we had the same. We weed it out and just put one of those in the apartment rather than having duplicates.
Except for little things like yearbooks, but you know we went to the same college, I've seen yearbooks. But that's that's a whole different thing. But that's that's basically what's happening here too. You know we have our stuff, the two circles are like our two apartments being combined. So the union is my stuff, his stuff, and then the joint stuff.
So let's do it without stuff but with elements for sets. Let's say your universal set has a star, an at symbol, an asterisk. hashtag or pound sign however you want to call it and a carrot okay let's say that set v contains star asterisk pound let's say that set w contains pound and carrot okay so now i draw the venn diagram We've got sets V and W. Sets V and W have the hashtag or pound sign in common, so that has to go in the middle here.
That's the intersection, the set containing that hashtag. Set V also contains star and asterisk. Set W also contains caret.
And then the universal set includes the at symbol. Okay, so if we want to talk about the union of V and W, we're taking everything from V, so everything from my apartment, everything from W, so everything from my husband's apartment, and we're putting it together. But notice that the hashtag only gets written down once.
Even though they both have the hashtag, hey, we're weeding out. We're only putting in, we have duplicates, we're just putting one of them in the apartment. We don't have room for two. And there you go. So the union has those four elements.
And that's it. So the union takes all of v's circle, all of w's circle, puts them together into one set. But doesn't duplicate what's in the intersection.
Now union and intersection do have a relationship to each other when you're trying to figure out how many elements there are. like in my example a minute ago I kept talking about how you do the Union you don't write down what's in common twice because this there's really no sense in doing that so when you're trying to figure out how many elements you should have in your Union you want to make sure you don't count those items twice okay so let's say you have So you have set A that contains alpha, beta, delta. Let's say you have set B that has beta, delta. Gamma. Alright, and we want to figure out how many items should be in the union.
Right now we've got number of items in set A is 3, number of items in set B is 3. So yeah, we know that when we do the union, we're putting sets A and B together. That's where the addition comes from. So we're taking the three items from set A plus the three items from set B and putting them together. The trouble is the union doesn't actually have that many items.
It doesn't have six because... We're not going to repeat stuff that overlaps, like the beta and the delta. We're not going to write it down twice.
But adding 3 and 3 counts them twice. So what we've got to do is we've got to look at how many is in the intersection. Let me see, beta and delta. And there's two items in the intersection. cardinal number of the intersection is 2. So we've got to subtract that out.
Subtract out the number of items in the intersection. So you subtract that 2 and that will give us the 4 that are truly in the union. There's something to be careful of when you're trying to figure out, well, how many should I have here?
Now, if the intersection happens to be zero, you know, if there's nothing in common, then yeah, just adding how many elements in A and elements in B together will give you your result for how many should be in the union. But that's in situations where there's nothing in the overlap. Alright, two more operations. Difference and Cartesian product. The difference is, you know, the difference you probably heard that word before because it's what we call the answer to a subtraction problem.
In set theory you're subtracting in a sense. You see that the symbol looks like a minus sign. What you're doing is you're taking whatever the first set has and you're subtracting what it has in common with the second set the second set can have additional elements that aren't common and they're not even going to come into play here all you're doing is taking out what it has in common with the first set and then seeing what's left over and that's it okay so let's say uh you have set set a contains a, b, c, d, e. Let's say you have set B contains d, e, f, g, h. Let's say you want to first find the difference of A and B.
So what you're going to do is you're going to write down set A and you're going to subtract from that you're going to look at what to a and B have in common a and B both have D E in common so you're going to subtract that out and see what you're left with that's it I know B also has elements FGH but that's irrelevant here. You're just taking A and subtracting out what it has in common with B. Now if you switch the order you're gonna find that difference is not commutative. In other words you're not going to get the same result because if you're doing the difference of B and A you have to start with set B and subtract out what it has in common with set A and see what's left over. So the operation is not commutative.
In other words, you switch the order, you're not going to necessarily get the same answer. It's possible to get the same answer, but not necessarily. It all depends on the sets.
That's how you find the difference. The Cartesian product looks like a multiplication sign, but they also call it A cross B. It's actually what you're doing is you're finding a set of ordered pairs. Whatever is listed first, those elements will be x coordinates, and whatever is listed second, those elements will be y coordinates, and then you'll have a set in the set braces of ordered pairs like x comma y. It's a little different than probably what you're used to.
Let's say you have a contains Apple and pear. Let's say that B contains red, blue, yellow. If you're going to do A cross B, Cartesian product of A and B. You're going to make ordered pairs where apple and pear are the x coordinates and red, blue, and yellow are the y coordinates.
So you'll have apple, red. You'll have apple, blue. Apple, yellow.
Moving on to pair. Pair, comma, red. Pair, comma, blue.
Pair, comma, yellow. Gosh, I wish it would quit doing that line. I don't know why it's doing that. The number of ordered pairs in A cross B. How many elements you get in that set is the number of elements in A times the number of elements in B.
So in this case, you had two elements in A, three elements in B, so you should have six ordered pairs in A cross B. And we do. We have 1, 2, 3, 4, 5, 6. And if you do B cross A, you will also have six ordered pairs.
Multiplication is commutative. 3 times 2 is also 6, but the Cartesian product is not commutative. Because when you switch the order, now B's elements are your X coordinates, A's elements are your Y coordinates.
So you do not end up with the same set when you switch the order like that. Now you got, now notice the X's and Y's are going to be swapped. Now you got red comma apple, red comma pear, red comma, or blue comma apple.
Sorry. Blue comma pear. You've got yellow comma apple and yellow comma pear. You swap the x and y.
Like here you have pair comma yellow. Down here you have yellow comma pair. So it's a whole different set. So cross product is not commutative. But you still end up with six ordered pairs.
When you do B cross A versus A cross B. Alright, then when you go to do problems in your homework, they're going to take these operations, and they're going to give you problems that combine a bunch of these operations into one problem. When you go to analyze those statements, use this order of operations, if you will.
Parentheses, always tackle those first, so take care of whatever's in parentheses. And whether you're inside parentheses or you've already worked your way out, do complements before all else, then do your other operations. Generally you can go left to right, but if you do the other operations slightly out of order, it's okay. Union, intersection, difference, Cartesian product, do all that stuff last. But if you have unions and intersections, it's really...
The parentheses are going to dictate what order to do things in. Beyond that, you can go left to right or something. It doesn't really matter.
But there shouldn't be any confusion because the parentheses are going to... keep it should keep it clear for you the problem. For example let's say I'm going to take this actually straight out of the current textbook on page 64 in questions 55 through 64 they say there's a universal set that contains one through eight the natural numbers one through eight And there's a subset of the universal set, set A, that contains 1, 2, 4, 5, 7. And there's a B that contains 2, 3, 5, and 6. Now you don't have to do this.
They did not provide a Venn diagram, so I'm going to draw one. It's terrible looking, isn't it? You do not have to do this.
The tablet's like leaning. Do you want, I'm going to erase that. That's going to drive me crazy now. You do not have to draw the Venn diagram, but I find it helpful to go ahead and draw it.
So Because I have a visual person. But again, you don't have to do this. Now some of the problems, they're going to give you a Venn diagram. So it doesn't hurt to be familiar with how to read it.
How to use it to do this. Sets A and B have 2 and 5 in common. Set A also has a 1, a 4, and a 7. Set B also has a 3 and a 6. So set A we got 1, 2, 4, 5, 7. Set B we got 2, 3, 5, 6. And then the universal set, 1, 2, 3, 4, 5, 6, 7, 8. Alright, one of the questions asked is find A union B in parentheses complement. Okay, do parentheses first. So first find A union B.
So that takes set A and set B and puts them together. You can either look at the roster form of the sets and just make sure you throw everything together, or you can come over here to the Venn diagram and just make sure everything the circles AB are included. Now that that takes care of what was in the parentheses, but working your way out now, it says do the complement of that.
So do the complement of the result you just got. So figure out what's left in the universal set that we've not included. And that's just the element 8. Because 8 union B contains 1, 2, 3, 4, 5, 6, 7. So 8 union B complement includes just 8. Now notice this is different than if you had been asked to do A complement union B complement. Two different questions there.
One of them had the parentheses and the complement outside. This one has no parentheses and has complements. So in this one you got to do the complements first.
You figure out, well, A complement is everything except 1, 2, 4, 5, 7. So it's 3, 6, and 8. B complement is everything except 2, 3, 5, 6. So it's 1, 4, 7, and 8. So then you put those together. A complement union B complement. Look at everything we just did.
A complement, B complement, and you're uniting them. So you're putting the 1, the 3, the 4, the 6, the 7, the 8, all of that into one set. Take care of what's in the parentheses first.
Take care of complements before you take care of everything else. Let me do one more of those. Let's see here. They can get really fun after a while.
There's one with a difference. Let's do the difference of A complement and B complement. We already know that A complement has 3, 6, and 8 from the example we just did.
And we know B Compliment has... 1, 4, 7, and 8. So when we go to do the difference of A complement and B complement, we write down A complement, which contains 3, 6, and 8, and we subtract or take out what it has in common with B complement, which is 8. And that just leaves us with the set containing 3 and 6. that's it okay well that's it for the operations that you're going to have to know how to do so far everything's been with two circles two subsets with the venn diagrams the operations you might find they like to kind of throw three sets at you too but you will not have to do venn diagrams yet for three sets we'll get to those next time but for now there you go so have fun and i look forward to seeing you the fun just continues when we do get to three sets and we start verifying whether they're equal and then this is setting us up to do some application problems basically it's not suspiciously like word problems so that's where we're heading all right well thank you for joining us and see you again soon