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## Statistics and probability

### Unit 7: Lesson 3

Basic set operations

# 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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• •   Z is the abbreviation for the german word "Zahlen." It's often used to represent the set of all integers - negative, 0 and positive.
• Could there be a number that is in a set AND also out of the set? •   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. :-)
• • 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!
• What are the representations of the set of all irrational numbers, of natural numbers, whole numbers and complex numbers? •  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.
• • 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)
• At that looks suspiciously like the image notation of a transformation in geometry,why is that? • How do you find the universal set of A = {1, 3, 5, Blue, `, Khan}, and B = {5, 9, 23, Vi hart, Lellow!}?   • 