## Sarah_Miller720 Group Title Suppose U = {1, 2, 3, 4, 5, 6, 7, 8} is the universal set and P = {1, 2, 3, 4}. What is P'? one year ago one year ago

1. Sarah_Miller720

i forgot how to do this stuff so i need some help :)

2. Sarah_Miller720

my choices. {5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4} cannot be determined

3. mivanov

well the first thing is that a universal set contains all sets including itself. for most set axiomatics you must know that such set does not exists, because of a paradox. Although there are some set axiomatics for which it exists

4. mivanov

Also for it be the universal set you can't say it has just 8 elements :D this is just plain not right

5. JayDS

Universal meaning "everything" so it must be everything inside P and also U.

6. mivanov

if we say that U is just a set(not the universal one) then we can say that P is a proper subset of U

7. mivanov

If a universal set exists, it's not a countable set. You should check out some set theory. It's cool.

8. JayDS

Universal meaning "everything" so it must be everything inside P and also U, excluding any reptitions.

9. JayDS

lol, u know ur theory stuff @mivanov , seems like u need to teach me some :)

10. mivanov

So you can't just list it's elements casually like that :D. And for most set theory axiomathics such set does not exists because of a simple paradox anyway.

11. Sarah_Miller720

So since i need everything then my answer would be{ 1, 2, 3, 4, 5, 6, 7, 8} right?

12. JayDS

yes u are right.

13. mivanov

thank you. Well there are 3 types of sets -> countably finite ex. {a,b,c} it has 3 elements because we can have bijective function f: {1,2,3} -> {a,b,c} that is for each element we can numerate it with a natural number. and the set {1,2,3} is the set of all numbers bellow or equal to 3. Second we have countably infinite sets. Imagine A is a set. If we can have a function f: N -> A such that it's bijective,(for each element of A there is unique element from N and the opposite is true as well) we call the set countably infinite. And last we have sets which are countably infinite. Example is the set of all subsets of the natural numbers. Such a set is called a power set. Namely for N it's writen P(N). The proofs of these statements aren't that long, if you are interested you can look around and read them.

14. mivanov

and if a universal set P exists, then we have that P is an element of P, also any elements composed of empty sets. Because the empty set is a subset of every set. Example we would have that {empty set} is an element of P, {{empty set}}, basically it has an infinite number of elements, and can't be comprehended easily. Also it leads to a paradox, because we have another axiom that tells us that for each property E(x) exists a set A for which the elements of A posses that property. It's called the Russel paradox.