 Welcome to our lecture series Math 3120, Transition to Advanced Mathematics for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. In this, the first video for lecture eight in our series, we're going to begin by talking about what a Venn diagram is. Many of us have seen these before, but the purpose of a Venn diagram is to visualize the interaction of sets based upon operations like intersections and unions and the like. The term Venn diagram itself is named after John Venn, who in around 1880 had developed these type of diagrams of overlapping circles. Although a similar idea was introduced by Leonard Euler, a very famous mathematician earlier than that. In this video, I did want to talk a little about Venn diagrams as we've been talking about sets, of course. A typical Venn diagram will look something like this right here, for which if you're having a Venn diagram for two sets, you have one circle that represents your first set A. You have a second set that represents, well, a second circle for the second set B right here. The fact that the overlap is supposed to indicate to the intersection. And this isn't always the case, but sometimes people will draw like a box or a rectangle around the circles. And this would represent like the universal set that the sets live inside of. All right, so let's look at a few illustrations of Venn diagrams. So if you want to illustrate the union of two sets, remember the union is every element that belongs to A or to B. So what you would do is you would color in the circle associated to A. You would then also color in the circle associated to B, like so kind of messy. Maybe we use a different color. And so the Venn diagram would then be both of those. We, of course, will just do it monochromatic here, just one color blue to take care of everything there. And so that's the union of those two sets. Now conversely, if we want to illustrate the intersection, the intersection between the two sets is the overlap. So you see that illustrated right here. Now another set construction that we've talked about before is the set difference. What if we want to consider all the elements of B that are not in A? Well, the Venn diagram that's exactly this portion kind of looks like some type of lunar shape of some kind. We want everything that's in B, but we take away the stuff that's also in A. So we take away their intersection and you're left with just this loon right here. That would be the set difference. Now remember the set difference is the same thing as A complement intersect B. And of course we could also color in this sector and we're over here. This would be the sector of A minus B and you can do some other ones. And now those give us some examples of Venn diagrams with two sets. It's also very common to talk about Venn diagrams with three sets, in which case you would illustrate that by three overlapping circles. One for set A, one for set B, one for set C, like so. And so notice that if we want to illustrate the set, the complement of A intersect B intersect with C. So notice this is not the same thing as A complement intersect B complement intersect C. There is an important thing there. We're taking the complement of A intersect B, which on our illustration here, A intersect B would be this thing right here. So we want everything that's not in A intersect B, but we also require that it be in C, like so. So if we want everything that's outside of this region, but is inside of C, you then see that the colored region is what then you get in that situation. This would be C take away A intersect B. So you could also write it using a set difference if you prefer, or you can do it with intersections and complements. Now it is useful to go the other way around that if you're given the diagram, can I tell you which set is represented by this illustration right here? It seems actually in some ways, similar to the picture we just talked about, there's something involved with the intersection of A and B, but now C has been removed. So it actually feels the other way around. My entire colored region lives inside of A intersect B, but you took away the triple intersection, A intersect B intersect, A intersect B intersect C. And so really we could represent this sector right here by taking A intersect B and then set difference C, or if you prefer A intersect B intersect C complement, that'll represent this region as well. And so this just gives you an illustration of how one can work with Venn diagrams for two sets or three sets. You can draw Venn diagrams for more than two sets. You typically don't draw circles anymore because the overlap just gets too hard to see. So you might draw some type of elongated blobby blob thing or something like, ooh, you know, something like that. You can do so. It's not as common as it is for Venn diagrams with three circles.