I need to prove that:
if A is an infinite set and x is some element such that x is not an element of A,
then (A union {x}) is equipotent to A.
The thing is I know it’s relatively easy to prove it with cardinal numbers, but thing is I cannot use any notion of cardinals because the exercise comes before that part of the course.
I tried three approaches:
- Considered the contrapositif: if A and A union {x} are not equivalent then A and A union {x} are finite. Then I tried several ways to prove it but none was satisfying, first assume they are countable so they are both equipotent to N and arrive at a contradiction. But then I arrive at uncountables and I freeze. I tried to utilize the theorem that there is always a countable set in an uncountable one, then I assume the cases where A (uncountable) – (countable set) = some set B. I then consider B finite arrive at a contradiction, then assume B is countable also arrive at a contradiction and then uncountable… and I freeze. I thought about doing the step again and proceed by induction since i will always have these three cases where 2 of them are absurd. But then what?
- I then decided to take a different approach, much more simple: take the definition of an infinite set specifically that if A is an infinite set then there is a proper subset S that is equipotent to it. Since A is a subset of A union {x} somehow or another i need to conclude it. The thing is the quantification: not any proper subset is equipotent to the parent’s…
- I then tried the other definition of an infinite set: that an infinite set has an injective function: A -> A ; that is injective but not surjective. I do not know however how to proceed about it then.
Any suggestions or help will be really appreciated. If it helps, the exercise is in the countable chapter of the course, and we finished: finite, infinite and countable. So its rather in the beginning of the course.