Intersection and union of sets

what I want to do in this video is familiarize ourselves with the notion of a set and also perform some operations on sets so a set is really just a collection of the distinct objects so for example I could have a set let's say let's call this set X and let's say and I'll deal with numbers right now but a set could contain anything it could take contain colors it could contain people that could contain other sets it could contain cars it could contain farm animals but the numbers will be easy to deal with just because well they're their numbers so let's say I have a set X and it has the distinct objects in it 3 the number 3 the number 12 the number 5 and the number 13 that right there is a set I could have another set let's call that set Y I didn't have to call it Y I could have called it a I could have called it Sal I could have called it a bunch of different things but I'll just call it Y and let's say that set Y it's it's a collection of the distinct objects the number 14 the number 15 the number 6 and the number 3 so fair enough those are just two set definitions the way that we typically do it in mathematics is we put these little curly brackets around the objects that are separated by commas now let's do some basic operations on sets and the first operation that I will do is called intersection and so we would say X we would say X intersect the intersection of x and y X intersect Y and the way that I think about this this is going to yield another set this is going to yield another set that contains the elements that are in both x and y so I often view I often view this intersection symbol right here as and so all of the things that are in X and in Y so what are those things going to be well let's look at both sets x and y so the number 3 is in set X is it in set y as well well sure it's both it's in both so it will be in the intersection of X & Y now the number 12 that's in set X but it isn't in Y so we're not going to include that the number 5 it's an X but it's not in Y and then we have the number 13 is in X but it's not in Y and so over here the intersection of x and y is the set that only has one object in it it only has the number 3 so we are done the intersection of x and y is 3 now another common operation on sets is Union so you could have the union of X x and y and the Union I often view or people often view as or so we're thinking about all of the elements that are in X or Y so in some ways you can kind of imagine that we're bringing these two sets together so this is going to be this is going to be and the key here is is that we care a set is a collection of distinct objects and the way we're conceptually right conceptualizing things right here this is the number 3 this isn't like somebody score on a test or the number of apples they have so there you could have multiple people with the same number of apples here we're talking about the object the number 3 so we can only have a 3 once but a 3 is in set is in X or Y so I'll put a 3 there a 12 is in X or Y a 5 is in X or Y the 13 is in X or Y and just to simplify things we really don't care about order if we're just talking about a set I've just put all the things that are in set X here and now let's see what we have to add from set Y so we haven't put a 14 yet so let's put a 14 we haven't put a 15 yet we haven't put the 6 yet and we already have a 3 in our set so there you go you have the union of x and y and one way to visualize sets and visualize intersections and unions and more complicated things is using a Venn diagram so let's let's say this whole box is you could that is the set of all numbers so that's all the numbers right over there we have set X set X I'll just draw as a circle right over here and I can even draw the elements of set X so you have 3 & 5 & 12 3 5 12 and 13 and then we can draw a set Y we can draw a set Y and notice I drew a little overlapping here because they overlap at 3 3 is an element in both set X and set Y but set Y also has the numbers 14 15 and 6 and so when we're talking about X intersect Y we're talking about where the two sets in overlap so we're talking about this region right over here and the only place that they overlap the way I've drawn it is at the number 3 so this is X intersect X intersect Y and then X Union Y is the combination of these two sets so X Union Y is literally everything everything right here that we are combining let's do one more example just so that we make sure we understand intersection and union so let's say that I have set a and set a is has the numbers 11 4 12 and 7 in it and I have set B I have set B and it has the numbers 13 4 12 10 and 3 in it so first of all let's think about what a a will be than a scholar let's think about what a intersect a intersect B is going to be equal to well it's the things that are in both sets so I have 11 here I don't have an 11 there so that doesn't that doesn't make the intersection I have a 4 here I also have a 4 here so 4 is an A and be it's an A&B so I'll put a 4 here the number 12 it's in a and B so I'll put a 12 here number Seven's only in a the number like 13 10 and 3 is only in B so we're done 4 and 12 the set of 4 and 12 is the intersection of sets a and B and we could even if we want to we could even label this as a new set we could say set C is the intersection of a and B and it's this set right over here now let's think about Union let's think about a a I want to do that in orange let's think about a union a union B what are all the elements that are in a or b a or b well we can just when we put all the elements in a 11 for 12 7 and then put the things in B that aren't already in a so C 13 we already put the 4 in the 12 a 10 and a 3 and I could write this in any order I want we don't care about order if we're thinking about a set so this right here is the Union