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.
prove the followings1- every subset of a finite set is finite2-.pdf
1. 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 XY
=> it implies that Y is a subset of X a
This set contain max n elements, hence the Y will be finite
