Basic set operations
Universal set and absolute complement
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- 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.
- If we're doing it as a Vien Diagram, the universe is usually depicted as some type of a rectangle right over here.
- It itself is a set and it is usually denoted with the capital U
- U 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 crazy because you're literally thinking of all possible things.
- Normally when people take 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 we'll talk about it in abstract terms right now.
- Let's say you have a subset of that universal set, set A.
- Set A contains 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.
- The way you could think about this is this is the set of all things in the universe that aren't in A.
- 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 the universal set minus A.
- Once again this is the capital U, this is not the union symbol over here.
- Or we could write this as U slash A.
- How do we represent that in the Vien Diagram?
- It would be all the stuff in U that is not in A.
- One way to think about it is as the relative complement of A that is in U
- but when you're taking the relative complement of something that is in the universal set
- you're really talking about the absolute complement.
- When people talk about the complement, that's what they're saying.
- What's the set of all the things in my universe that are not in A?
- Let's make things a little bit more concrete by talking about sets of numbers.
- Once again, we could have been talking about sets of TV personalities, or animals, or whatever it might be
- but numbers are a nice simple thing to deal with
- and let's say that our universe that we care about is the set of integers.
- Our universe is the set of integers.
- I'll write captial U is the set of integers.
- This is a little bit on the side, but the notation for the set of integers tends a bold Z.
- It's Z for "zol" from German for integer.
- The bold is this kind of weird looking "blackboard bold".
- it's what mathematicians use for different types of sets of numbers.
- I'll do a little on the side here.
- They might write R like this for the set of real numbers.
- They'll write a Q in that blackboard bold font (it looks something like this), this is the set of rational numbers.
- You might say, "Why Q for rational?"
- Well, there's a couple of reasons:
- One, the R is already taken up
- and Q for quotient, a rational number can be expressed as a quotient of integers
- and we just saw the Z for zol or integers, the set of all integers.
- So our universal set, the universe that we care about right now is integers
- and let's define a subset of it, let's call that subset C (a letter that I haven't been using a lot).
- The set C is equal to -5, 0, and +7.
- I'm obviously not drawing it to scale
- the set of all integers is infinite while the set C is a finite set
- but just to draw it, that is our set C right there.
- And let's think about what is a member of C and what is not a member of C.
- We know that -5 is a member of our set C.
- This little symbol right here denotes membership, it looks a lot like the Greek letter epsilon, but it is not
- it just means membership of a set.
- We know that 0 is a member of our set, and 7 is a member of our set.
- We also know some other things.
- We know that the number -8 is not a member of our set.
- We know that the number 53 is not a member of our set.
- 53 is sitting some place out here.
- We know the number 42 is not a member of our set, 42 might be sitting some place 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 relative complement C.
- These are all equivalent notation.
- What is this in our diagram? It's all the stuff outside of our set C.
- And now we know that -5 is a member of C, so it can't be a member of C complement
- so -5 is not a member of C complement.
- 0 is not a member of C complement. 0 sits in C, not in C complement.
- 53 is a member of C complement. It's in the universe, but outside of C.
- 42 is a member of C complement.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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