Allow me to point out that it is usual to define .
This proof also uses the idea of characteristic function:
Note that .
Now define .
Now it up to you to show that is a bijection.
Given two sets and , let be the set of all functions . Prove that for any set , .
Here, . In other words, we want to prove that ?
So is the set of all functions . It does not say anything about the sets being countable/uncountable, or finite/infinite. Since any element maps to either of two values, then using the characteristic function, the total number of functions is if is countably infinite or finite. But if is uncountable/infinite, then ?
Well, if you're going to prove , then shouldn't A be a finite set with n elements?
It's a matter of combinatorics. Any element in A can be mapped to 0 or 1, so there are exactly two choices for it. Since there are n elements in A, there are ways of mapping A to {0,1}; hence there are functions from A to {0,1}.
This is a little bit off topic but you might it interesting we define to be where and . This is what Plato said. Thus, there is no need to write because in fact .
This might look strange but this is how we define natural numbers. We define . We define . We define . We define . We would like to say natural numbers are repetitions of this form. However, this is not a formal enough statement. So this is what we do. Let be a set and define . Thus is follows . We cannot say "and so on" because that is not a formal term. To do this we define an inductive set to be a set such that and if then . One of the axioms we place into Set Theory is called Axiom of Infinity which says "an inductive set exists", the reason we do this is because using the simpler axioms we cannot construct inductive sets and we would like to make set theory more interesting by allowing infinite sets. Finally, since there is an inductive set define . This set are called the natural numbers and this is a purely formal construction of the naturals.
And that is precisely why we can't have half integers contained in , because of the principle of induction?
Moreover, the natural numbers never repeat (e.g. we cannot have ? What is the difference between formal difference and actual difference (e.g. we have and )?
I do not know what you mean by this.
I do not know what you mean by this. Do define we find need to define what it means for arbitrary integers.Moreover, the natural numbers never repeat (e.g. we cannot have ? What is the difference between formal difference and actual difference (e.g. we have and )?