How Do You Prove Sets Have Cardinality Aleph-Nought?

[dhillon]

URGENT HELP PLEASEEEE, a question on set theory

Homework Statement

the question is:

Prove that these sets have cardinality aleph-nought:(there is two 2 prove)

(a) {1/(2^k) : k∈ℕ}

(b) {x∈ℤ : x >= -5}


im not sure how to work this out, please help on this, i did ask on a previous thread how to prove cardinality of a statement, thanks for your help guys

 

[Mark44]

Homework Statement

the question is:

Prove that these sets have cardinality aleph-nought:(there is two 2 prove)

(a) {1/(2^k) : k∈ℕ}

(b) {x∈ℤ : x >= -5}


im not sure how to work this out, please help on this, i did ask on a previous thread how to prove cardinality of a statement, thanks for your help guys

I'm using IE8, which doesn't display some symbols. For a, I'm guessing that it says that k is a positive integer. I have no idea what the two symbols after x are in the b part.

To show that the cardinality of a set is Aleph-nought, show that there is a one-to-one pairing between the elements in the set and the positive integers.

 

[dhillon]

hey thanks for trying to help, I truthfully have no idea on how to do this,

part a) k is an element of natural number e.g 1,2,3,4...
part b) x is an element of integers e.g -3,-2,0,1,2,3...

if this has helped please let me know

 

[Mark44]

hey thanks for trying to help, I truthfully have no idea on how to do this,

Did you miss the second paragraph in my post?

part a) k is an element of natural number e.g 1,2,3,4...
part b) x is an element of integers e.g -3,-2,0,1,2,3...

In part b, the set is {-5, -4, -3, ..., 0, 1, 2, 3, ...}

 

[dhillon]

oh ok sorry i missed that, i'll try, thanks for the help, i was trying since morning but I am not sure that's the thing, i'll keep trying though, do you know how to work this out by any chance? because I am soo stuck :( , thank you

 

[Mark44]

Yes, I know how to do them. Neither one requires much work. The first one is almost obvious.

For a, write the set in expanded form, starting with the first member and continuing for 5 or 6 members. Show that each member in this set can be associated with one of the numbers in the set {1, 2, 3, 4, ...} and be able to show the pairing for an arbitrary member of your first set.

It's very similar for the b part.