Hello,
How come the cardinality of set can be presented as ,wre denotes the cardinality of set ? How can it be justified?
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
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.
So I've written to my tutor asking for help and here's what he had to say:
Can somebody please explain to me how come ? From what I understand for set is a number/cardinal number that defines the cardinality of the set, but I didn't know that the cardinal number has the same cardinality as the set it corresponds to, which what the tutor wrote to me suggests.Since you know that and , then , which means that (because ).