- #1
mahler1
- 222
- 0
1. Homework Statement .
Let A and B be disjoint sets, A infinite and B enumerable. Prove that there exists a bijection between AuB and A.
3. The Attempt at a Solution .
I have an idea of how to prove this statement, but I got stuck in the middle, so here is what I've done:
There are just two cases to consider:
1)A is enumerable or 2)A is uncountable.
I think that the first case is pretty simple, since if A and B are both enumerable and disjoint sets, then, AuB is also enumerable. By definition of enumerable, there exist f and g both bijective functions from N to A, and from AuB to N respectively. If I define h=fog, this function is bijective because it is composition of bijective functions and it goes from the set AuB to the set A. In conclusion, h is the function that satisfies the required conditions.
I am having some difficulty in constructing the desired function when A is uncountable. If A is uncountable, then it has a proper subset which is countable. Let's call that set S. Using properties of cardinality, I can prove that A-S is still uncountable. And the set A is exactly the union of A-S and S.
The idea I had to construct the bijection was defining the function in this way:
f restricted to the set B is the composition hog where h is a bijective function with N as its domain and S as its codomain and g is a bijective function with B as its domain and N as its codomain. Now I need to define f restricted to the set A, and here is where I get stuck. I mean, can I prove that there is a bijection from A to its proper uncountable subset A-S?. If this was true, then it would we very easy to prove that f is a bijection from AuB to the set A=(A-S)uS.
Or maybe there is an easier way to solve the problem, without having to consider separated cases but this was all I could think of.
Let A and B be disjoint sets, A infinite and B enumerable. Prove that there exists a bijection between AuB and A.
3. The Attempt at a Solution .
I have an idea of how to prove this statement, but I got stuck in the middle, so here is what I've done:
There are just two cases to consider:
1)A is enumerable or 2)A is uncountable.
I think that the first case is pretty simple, since if A and B are both enumerable and disjoint sets, then, AuB is also enumerable. By definition of enumerable, there exist f and g both bijective functions from N to A, and from AuB to N respectively. If I define h=fog, this function is bijective because it is composition of bijective functions and it goes from the set AuB to the set A. In conclusion, h is the function that satisfies the required conditions.
I am having some difficulty in constructing the desired function when A is uncountable. If A is uncountable, then it has a proper subset which is countable. Let's call that set S. Using properties of cardinality, I can prove that A-S is still uncountable. And the set A is exactly the union of A-S and S.
The idea I had to construct the bijection was defining the function in this way:
f restricted to the set B is the composition hog where h is a bijective function with N as its domain and S as its codomain and g is a bijective function with B as its domain and N as its codomain. Now I need to define f restricted to the set A, and here is where I get stuck. I mean, can I prove that there is a bijection from A to its proper uncountable subset A-S?. If this was true, then it would we very easy to prove that f is a bijection from AuB to the set A=(A-S)uS.
Or maybe there is an easier way to solve the problem, without having to consider separated cases but this was all I could think of.