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Universal set and absolute complement

Sal moves onto more challenging set ideas and notation like the universal set and absolute complement. Created by Sal Khan.

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  • leaf green style avatar for user FinallyGoodAtMath
    What is the German word that Z stands for?
    (42 votes)
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  • leaf blue style avatar for user David Elijah de Siqueira Campos McLaughlin
    Could there be a number that is in a set AND also out of the set?
    (27 votes)
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    • male robot hal style avatar for user Ben Willetts
      Assuming you mean "can there be a number that is simultaneously in a set and not in the set?" then no, that's what's called a paradox.

      It's actually quite complicated to set one up -- an example would be "the set of all sets that do not contain themselves". This set must simultaneously contain itself and not contain itself. We resolve the paradox by saying that such a set cannot be constructed, and leave it at that. :-)
      (75 votes)
  • spunky sam green style avatar for user John
    does universal set contains itself?
    (21 votes)
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    • leaf red style avatar for user Philosoraptor
      This is going to be untrue in many cases where the domain is restricted and unclear in certain paradoxical cases. For example, if we restrict our domain/universe to only sets, and further to only sets that do not contain themselves (the set of all red things does not contain itself, because sets are abstract objects and therefore are not colored) - then if it contains itself, it doesn't contain itself (because it is the set of all sets that do not contain themselves). The paradox comes in when you notice that because it is the set of all sets that do not contain themselves, if it doesn't contain itself, then it does contain itself - and round and round we go!
      (7 votes)
  • aqualine ultimate style avatar for user QUIDES
    What are the representations of the set of all irrational numbers, of natural numbers, whole numbers and complex numbers?
    (3 votes)
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    • male robot hal style avatar for user Yamanqui García Rosales
      The natural numbers are the numbers you use for counting things (with or without 0, there is no consensus about it), it can be represented as ℕ = {1, 2, 3, 4 ⋯ }

      The whole numbers are usually called "integers" and includes all the natural numbers, plus their negatives (and 0), it's represented as ℤ = {⋯ -3, -2, -1, 0, 1, 2, 3 ⋯}

      Rational numbers are all the numbers that can be represented as a fraction of two integers, it's represented as 𝐐 = { a/b ; a∈ℤ, b∈ℤ, b≠0 }

      Irrational numbers are all the real numbers that cannot be represented as a fraction of two integers. There is no standard notation for this set, but you can get it by subtracting form the set of all the real numbers () the rational numbers: ℝ\𝐐

      Complex numbers are all the numbers that have a real and an imaginary part, where the coefficient of the real and imaginary part is a member of the real numbers, it's represented as ℂ = {a + ℹb ; a∈ℝ, b∈ℝ}

      From this definitions you can see that the most general set is the set of the complex numbers, that include all other sets. Next one is the set of the real numbers, that are formed by the union of the rational and the irrational numbers. The rational numbers further include the set of the integers, and finally the set of the natural numbers is the smallest of them all.
      (26 votes)
  • aqualine ultimate style avatar for user Vinay Manjunath
    Can any one explain about absolute complement ?
    (2 votes)
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    • female robot grace style avatar for user Dhanishthaghosh
      Absolute complement of a set is said to be the set which contains all the elements of the universal set other than the prior set. For example, if U is the universal set of all numbers starting from 1 to 100. U={1,2,3,4,5,6,7,.......,99,100} Suppose Dis a set which contains numbers from 1 to 50. A={1,2,3,4,5,.....,49,50} Then absolute complement of A is numbers from 51 to 100. A'={51,52,53,54,......,99,100}.
      I hope you got it well. Thanks!
      (1 vote)
  • male robot hal style avatar for user RN
    At that looks suspiciously like the image notation of a transformation in geometry,why is that?
    (4 votes)
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  • orange juice squid orange style avatar for user Ramana
    How do you find the universal set of A = {1, 3, 5, Blue, `, Khan}, and B = {5, 9, 23, Vi hart, Lellow!}?
    (0 votes)
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    • aqualine tree style avatar for user Ted Fischer
      You cannot extrapolate from a given set to a universal set. The universal set needs to be specified or implied from the start. For example, A = {1, 3, 5} could be taken from the universal set of:
      (1) The counting numbers.
      (2) The odd integers.
      (3) The rational numbers.
      (4) The real numbers.
      ...and so on...
      (7 votes)
  • blobby green style avatar for user jayashree635
    If universal set contains everything in this univers,why doesn't it contains the elements of set A?
    (2 votes)
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  • duskpin ultimate style avatar for user ★彡 ✦StarLight✦ 彡★
    Does Sal have a video on three sets and how to shade them? Thanks!
    (2 votes)
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  • marcimus pink style avatar for user tanmay deb
    Is it compalsary to put u in the venn diagram
    (2 votes)
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Video transcript

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 for 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-- 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 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 we'll talk about in abstract terms right now. Now, let's say you have a subset of that universal set, set A. And so set A literally 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. And the way you could think about this is this is the set of all things in the universe 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 minus 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, U slash A. So how do we represent that in the Venn diagram? Well, it would be all the stuff in U 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 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. Or when people just 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. 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 that we care about right over here is the set of integers. So our universe is the set of integers. So I'll just 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 for Zahlen, from German, for apparently integer. And the bold is this kind of weird looking- they call it blackboard bold. And it's what mathematicians use for different types of sets of numbers. In fact, I'll do a little aside here to do that. So for example, they'll write R like this for the set of 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 a 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 two integers. And we just saw you have your Z for Zahlen, 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-- I don't know. Let me use a 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 to draw it, 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 C. This little symbol right here, this denotes membership. It looks a lot like the Greek letter epsilon, but it is not the Greek letter epsilon. This just literally means membership of a set. We know that 0 is a member of our set. We know that 7 is a member of our set. Now, we also know some other things. We know that the number negative 8 is not a member of our set. We know that the number 53 is not a member of our set. And 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? Well, it's all this stuff outside of our set C right over here. And now, all of a sudden, we know that negative 5 is 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, not in C complement. I could say 53-- 53 is a member of C complement. It's outside of C. It's 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.