Question

Can someone please check my answers and guide me with the 1st choice?
Multiple choice with more than one answer possible.
Which of the following are true?
1) All uncountable sets have the same cardinality. (<-- I think this is my problem)
2) Any set of real number is uncountable (FALSE b/c the set of integers are real and they are countable, right?)
3) The rational numbers form a countable set. (TRUE)
4) The power set of any infinite set is countable. (I thought this was FALSE b/c infinite sets can be either countable OR uncountable, correct?)
5) The real numbers form a countable set. (FALSE, correct?)

Answer 1

Lets think about this..
Let A={3,5}
the powerset(A)={emptyset, {3},{5},{3,5}}
|A|=2
|powerset(A)|=4
and 4>2
so A and the powerset(A) do not have the same cardinality
Now I know these sets are not infinite...
But let's think about an infinite set.
Like the real numbers...
\[|\mathbb{R}| <|powerset( \mathbb{R})|\]

Answer 2

those are both uncountably infinite sets but you see the sizes are different

Answer 3

That makes sense, thank you. My other ones must be incorrect then?
My professor said my thoughts on #3 were correct but that there was another true answer. ??

Answer 4

@freckles - This was his reply to me: Cantor's argument shows that any real interval is uncountable, but you can come up with sets of real numbers that are countable. The integers, after all, form a set of real numbers.

Your answer about the rational numbers is fine.

There's one more true statement in the list. Remember that we did show that for any set S, S and its power set have different cardinalities; that holds true even when S is uncountable.

Answer 5

So let's look at number 2) that is false because
the integers is a set of real numbers that are countable
well and also the set that is {4,6} is a set of real numbers and it is also countable
so your answer looks good to number 2
3) is true too the rational numbers can be listed
4) why did you say 4 was false?

Answer 6

the real numbers are not countable but are infinite
so how can the power set of the real numbers be countable?

Answer 7

right, so doesn't your explanation make #4 false? The power set of any infinite set is countable. Not always, right? so isn't that false?

Answer 8

Yeah you are right

Answer 9

I don't know why I was arguing it for to be false when you said it was false.

Answer 10

Well I'm missing something somewhere....There is at least one more true answer.

Answer 11

1) False because |R|<|power(R)| <--your teach even just said this
2) False because the integers is a set of real numbers that are countable
3) true
4) false because |power(R)| is not countable
5) False the real numbers don't form a countable set but a subset of the real numbers like the integers do (but not all subsets of the real numbers will)

Answer 12

so maybe I'm missing something too

Answer 13

But I don't think I am
we have come up with an example of each them being false except 3

Answer 14

like each of these are word for for?

Answer 15

word for word*

Answer 16

The power set of any "countably" infinite set is countable would be true

Answer 17

I have to re-open the assignment to check word for word....I can do that now and only have an hour to complete it. So, since we've got most of this worked out, I'll open it and check the options word for word. I'll be back in a few moments and let you know. :)

Answer 18

k

Answer 19

@freckles here's a screen print of the question (attached file)

Answer 20

you know I think I was wrong about the power set question

Answer 21

you know we just said and your teacher said the size of a set and its power set is different

Answer 22

I did type it wrong too! It says the power set of any infinite set is UNcountable.

Answer 23

oh yeah

Answer 24

which is true

Answer 25

Like you know for some reason I thought the power set of the integers would be countable
but I don't think it is
we can use the cantor's diagonalization method or whatever

Answer 26

well have fun
i guess you are done with your assignment

Answer 27

Thank you so much for your help and walking me through this! I'll submit it now and let you know how we did! :)

Answer 28

@freckles - That was it! Thanks again!!

Answer 29

np

Answer 30

You can have a medal to. You did all the work and you were right about everyone of them almost.

Answer 31

Awww thanks :)