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## Statistics and probability

# 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.

## Want to join the conversation?

- What is the German word that
**Z**stands for?(42 votes)**Z**is the abbreviation for the german word "Zahlen." It's often used to represent the set of all integers - negative, 0 and positive.(101 votes)

- Could there be a number that is in a set
**AND**also out of the set?(27 votes)- 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)

- does universal set contains itself?(21 votes)
- 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)

- What are the representations of the set of all irrational numbers, of natural numbers, whole numbers and complex numbers?(3 votes)
- 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 (`ℝ`

) 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)

- Can any one explain about absolute complement ?(2 votes)
- 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)

- At1:19that looks suspiciously like the image notation of a transformation in geometry,why is that?(4 votes)
- Cause it's the same symbol. :)(1 vote)

- How do you find the universal set of A = {1, 3, 5, Blue, `, Khan}, and B = {5, 9, 23, Vi hart, Lellow!}?(0 votes)
- 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)

- If universal set contains everything in this univers,why doesn't it contains the elements of set A?(2 votes)
- Universe = set A (A) U (union) compliment of A (A').

Universe = A U A'(2 votes)

- Does Sal have a video on three sets and how to shade them? Thanks!(2 votes)
- Is it compalsary to put u in the venn diagram(2 votes)

## 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.