Universal set and absolute complement

what I want to do in this video is introduce the idea of a universal set or the universe that we care about and also the idea of a complement or an absolute complement and so if we're doing it as a Venn diagram the universe is usually depicted as some type of a rectangle right over here and it itself is a set and it usually is denoted with the capital u you for universe not to be confused with the union set notation and you could say that the universe is all possible things that could be in a set including farm animals and kitchen utensils and emotions and types of Italian food or even types of food but then that just becomes somewhat somewhat crazy because you're literally thinking of all possible things normally when people talk about a universal set they're talking about a universe of things that they care about so the set of all people or the set of all real numbers or the set of all countries whatever the discussion is being focused on but will talk about in abstract terms right now and let's say you have a subset of that universal set so let's say you have a subset of that universal set set a and so set a is literally contains everything everything that I have just shaded in what we're going to talk about now is the idea of a complement or the absolute complement of a + 1 and the way you could think about this is this is the set of all things in the universe in universe that aren't that aren't that aren't in a and we've already looked at ways of expressing this the set of all things in the universe that aren't in a we could also write as a universal set - a once again this is a capital u this is not the union symbol right over here or we could literally write this as U and then we write that little slash looking thing you slash a so how do we represent that in the Venn diagram it'll be all the stuff in you it would be all the stuff in you that is not in a one way to think about it you could think about it as the relative complement of a that is in you but when you're taking the relative complement of something that is in the universe Versalles set you're really talking about the absolute complement or when people just talk about the compliment that's what they're saying what's the set of all the things in my universe that are not that are not in a now let's make things a little bit more concrete by talking about sets of numbers once again our sets we could have been talking about sets of TV personalities or sets of animals or whatever it might be but numbers are a nice simple thing to deal with and let's say that our universe our universe that we care about right over here is the set of integers so our universe is a set of integers so I'll write u capital u is equal to the set of integers and this is a little bit of an aside but the notation for the set of integers tends to be a bold Z and it's z fours all from German for apparently integer and the bold is this kind of weird-looking they call it blackboard bold and so what mathematicians use for different types of sets of numbers and in fact I'll do a little aside here to do that so for example they might say they'll write R like this for the set of real numbers real numbers they'll write a Q in that blackboard bold font so it looks something like this they'll write the Q it might look something like this this would be the set of rational numbers and you might say why Q for rational well there's a couple of reasons when the R is already taken up and Q for quotient a rational number can be expressed as a quotient of two integers and we just saw you have your z fours all fours all or integers the set the set of all integers so our universal set the the universe that we care about right now is integers and let's define let's define a subset of it let's call that subset I don't know I've been let me put a user letter that I haven't been using a lot let's call it C the set C let's say it's equal to negative 5 0 and positive 7 and I'm obviously not drawing it to scale the set of all integers is infinite while the set C is a finite set but I'll just kind of just draw it all that's our set C right over there and let's think about what is a member of C and what is not a member of C so we know that negative 5 is a member of our set see this little symbol right here this denotes membership membership it looks a lot like the Greek letter epsilon but it is not the Greek letter epsilon this is just literally means membership of a set we know that 0 is a member of set of so we know that 0 is a member of our set we know that 7 is a member is a member of our set now we also know some other things we know that the number negative 8 is not is not a member of our set we know that the number 53 is not a member not a member of our set 53 is sitting someplace out here we know the number 42 is not a member of our set 42 might be sitting someplace out there now let's think about C complement or the complement of C C complement which is the same thing as our universe minus C which is the same thing as universe or you could say the relative complement of C in our universe these are all equivalent notation what is this first of all in our Venn diagram what's all this stuff outside of our set outside of our set C right over here and now suddenly we know that negative 5 negative 5 it was a member of C so it can't be a member of C complement so negative 5 is not a member of C complement 0 is not a member of C complement 0 sits in C naught and C complement negative or I could say 53 53 is a member of C complement it's outside it's outside of seats in the universe but outside of C 42 is a member of C complement so anyway hopefully that helps clear things up a little bit