(a) $[1,3)$, $\mathfrak{c}$
(b) $Z$, $\aleph_{0}$
(c)$R \times R$,
(d) $R \cap Z$,
(e) $\{ 2^{-k} : k \in \mathbb{N} \}$
I understand that aleph null means that it is infinite and that c means the set is finite like (0,1). What I am confused about is letters c through e. what is the cross product of rational numbers is it finite? The intersection of rational and integers is confusing because I do not understand where it starts. For e I am confused as well as where it starts.
On
$\mathbb Q\times \mathbb Q $ has the cardinality $\aleph_0$, it is possible to show that $|\mathbb Q\times \mathbb Q|=|\mathbb Q|=\aleph_0$.
Now $(\mathbb Q \cap\mathbb Z)\subset \mathbb Q$ so it has the cardinality $\aleph_0$. You can resolve (e) in the same way.
On
Okay. Even more basic. What does it mean when we count? What does it mean to say there are 4 Beatles, or there are the same number of Beatles as points on the compas.
Well, this means there is a 1-1 corespondence between these sets and the set {1,2,3,4}. There are 4 beatles because we can map 1 <=> George <=> north; 2 <=>paul <=> east; 3 <=>ringo <=>south and 4 <=>john <=>west.
This is what we mean when we say {Beatles}, {1,2,3,4},and {compas points} have the same cardinality and that cardinality is "4".
Now obviously some sets have have a different number of elements/cardinality. |{Beatles}| =4 but |{colors of rainbow}|= 6. How can these corespond.
Well there can't be a 1-1 mapping, obviously. But we can map {Beatles} into {colors}. Ringo <=>red,george <=>green,paul <=>purple, and John<=>orange. But this isn't onto because nothing is mapped onto blue or onto yellow. It should be obvious why this is impossible.
Likewise it's impossible to map {colors} into {Beatles} because after the first four mappings we run out of options to map the final two color into.
So if we can map A into B but we can't map A onto B (or equivalently we can't map B into A) we say the cardinality of A is less than B.
|{Beatles}| < |{colors of rainbow}| and 4 < 6. Big surprise.
Okay. That is what counting means, that is what cardinality means and that is what sets being bigger or smaller than others mean.
This is easy. A bit... weird pedantic and obtuse. But easy.
But some sets aren't finite. Well, obviously an infinite set is bigger than a finite set. We can map {1,2,3....,50} into N={1,2,3....} but we can't map N into {1,2,3...50}. So infinity=|N|>|{1,2,...50}|=50.
And sometimes infinite sets have the same cardinality. k <=>2k is a one to one mapping between the even numbers and the whole numbers and +z,0 <=> 2z+1 with -z <=>2z is a one to one mapping from the integers Z to the natural numbers N.
So Z, N, even numbers all have the same infinite cardinality. We say the sets are countable and we call the cardinality AlephNull.
But Cantor showed we can map N into the real numbers but we can not map N onto the real numbers.
So the real numbers have a larger cardinality than the cardinality of the Natural numbers.
We call this cardinality continuum.
And we have a hierarchy of cardinalities. 0 <1 <2 <....... < AlephNull < continuum.
Sidenote: how did Cantor show we can map N into but not onto the reals? He assumed if we could map N to (0,.99999....=1] we count make a 1-1 list between n <=>.asdfggh... between o natural number and a decimal expansion. If we consider a decimal where the i-th digit was different than the i-th digit of i-th term on the list. That decimal is different than the i-th item on the list. Do this so that every digit is different than the corresponding digit of the corresponding item on the list. The resulting decimal is a decimal between 0 and 1 and different then every decimal on the list. That means it was on the list. That means a list is impossible. We can't map N into (0,1].
So |(0,1]| > |N|.
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Okay. Basics.
1)for finite sets cardinality is how many elements the set has.
2) some sets are infinite so some types of cardinality describe what type of infinite the sets are.
3) sets have the same cardinality if they can be put in 1 to 1 order with each other.
4) Some infinite sets can not be put in 1 to 1 order. Example the counting numbers 1,2,3... and the set [0,1] (all the real numbers between 0 and 1) have different cardinality. [0,1] is a "bigger" infinity than 1,2,3.... (We use Cantor's diagonal for that.)
5)So we have, to begin with, 3 types of cardinality: finite; the cardinality of the natural numbers-- we call such sets "countable" and call the cardinality AlephNull (sorry, I can't do symbols); The cardinality of [0,1]-- we call these sets "uncountable" and we call the cardinality c (I'll call it "continuum" because I can't do symbols).
6) if $A\subset B $ then $|A| \le |B|$
7) Here are some sets that are the same cardinality as the natural number; AlephNull:
Z-- the set of integers... because 2n <-> +n and 2n-1 <-> is 1-1 correspondence.
ZxZ the cross product of integers. 1 <->1,1 2<-> 2,1 3 <-> 1,2 4 <->3,1 5 <-> 2,2 etc... is 1-1 corespondence between N and NxN and as Z~N then Z~ZxZ.
The rational numbers Q. a/b <-> a,b maps Q to a subset of ZxN~N.
8) Here are some sets with the cardinality of [0,1]; continuum.
(0,1). The reals R. RxR.
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So... your questions.
a) [1,3). We can stretch and switch this to (0,1] in a 1-1 mapping. So continuum.
b) we can map the negative and zero integers to the positive even whole numbers and the positive integers to the odd whole numbers so there is a 1-1 mapping between the integers and the natural numbers. AlephNull.
c)let r and s be real numbers in decimal form. Let t be a real number in decimal form where the odd term digits are the digits of r and the even term digits are the digits of s. r x s <=> t is a 1-1 mapping R x R <=> R. Continuum.
d) $R \cap Z = Z $. So AlephNull
e) n <-> $2 {-n} $ is 1 to 1 corespondence so AlephNull.
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Postscript. (0,1) $\subset (0,1] \subset [0,1] \subset (-1,2) $ so these all have cardinality continuum.
x<=>x/|x+1| is a 1-1 mapping from R to (-1,1) so R has cardinality continuum.
Some pointers:
If there are infinitely many members in a set, and you can bring them in bijection with $\mathbb{N}$, $\mathbb{Z}$, or $\mathbb{Q}$, that set is “countable”, and has cardinality $\aleph_0$.
If there are infinitely many members in a set, and you can bring them in bijection with $\mathbb{R}$, that set has cardinality $\mathfrak{c}$.
If there is a surjection from your set to $\mathbb{R}$, it must be “bigger than” $\mathbb{R}$, so it’s at least of cardinality $\mathfrak{c}$.
If your set is infinite, and it’s a subset of some set with cardinality $\aleph_0$, it must have cardinality $\aleph_0$ itself, since that’s the smallest infinite cardinal.
These should help you solve (c), (d), and (e).