Hello,

How come the cardinality of set can be presented as ,wre denotes the cardinality of set ? How can it be justified?

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- Dec 30th 2012, 11:22 AMMachinePL1993Cardinal Exponentation
Hello,

How come the cardinality of set can be presented as ,wre denotes the cardinality of set ? How can it be justified? - Dec 30th 2012, 12:16 PMPlatoRe: Cardinal Exponentation

The justification for using that notation, , for the set of all functions is a generalization from the finite case.

If both are finite sets, where then**any function**

is just having the properties:

.

So the function has exactly distinct pairs.

Each of those pairs has possible second terms.

Thus there are or possible functions from - Dec 30th 2012, 03:11 PMMachinePL1993Re: Cardinal Exponentation
What if we are dealing with infinite sets?For example how would we denote the cardinality of a set of all functions ?

Can we also do it like this?

- Dec 30th 2012, 03:21 PMPlatoRe: Cardinal Exponentation
- Dec 30th 2012, 03:29 PMMachinePL1993Re: Cardinal Exponentation
How does relate to the fact that ? I know how to prove this property, but I have no idea how it relates to exponentation.

- Dec 30th 2012, 03:40 PMPlatoRe: Cardinal Exponentation
- Dec 30th 2012, 04:24 PMHallsofIvyRe: Cardinal Exponentation
If you know how to prove it, you should at least know what it looks like for

**finite**sets! If |A| has n elements, then the A has a total of subsets.

For example, if |A|= 0, then A is the empty set: {}. Its only subset is {} itself so it has subset. If |A|= 1, then it has 1 member. Call that member "x". Then A= {x} so the two subsets are {} and {x} itself. If |A|= 2, then, say, A= {1, 2} so that its subsets are {}, {1}, {2}, {1,2} a total of subsets. If |A|= 3, we can represent A as {1, 2, 3}. Its subsets are {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, a total of subsets.

One can prove that if |A|= n then A has subsets by induction on n.

That is where the notation comes from. - Dec 30th 2012, 05:19 PMMachinePL1993Re: Cardinal Exponentation
So I've written to my tutor asking for help and here's what he had to say:

Quote:

Since you know that and , then , which means that (because ).