prove the followings 1 every subset of a finite set is finit

prove the followings:

1- every subset of a finite set is finite

2- if s is denumerable, then s is equinumerous with a proper subset of itself

Solution

Post one more question to get the answer to the second problem. Thanks

Proof using induction:

1) Base Case (n=1)

There is only one element in the finite set, then we can write it as

X \\ a = Phi, since there won\'t be any element in the set if we removing phi, hence the phi is a finite set since it contains only one element and this is an empty element

2) Let us assume that the solution is true for (n=k), i.e. for an finite set with k elements, every subset of that set will be finite

3) Now we need to prove our hypothesis for the set containing (k+1) elements

Let us assume that X has (k+1) elements if Y is a subset of X, then it will have atmost k+1 elements, which is equal to X

Since X is finite, therefore Y will also be finite

Otherwise, there exists an element that belong to X but not to Y

a belongs to X\\Y

=> it implies that Y is a subset of X\\ a

This set contain max n elements, hence the Y will be finite