Infinite sets statements equivalence

[mahler1]

Homework Statement

Let A be a set, prove that the following statements are equivalent:

1) A is infinite
2) For every x in A, there exists a bijective function f from A to A\{x}.
3) For every {x1,...,xn} in A, there exists a bijective function from A to A\{x1,...xn}

Relevant equations

The first and only thing that comes to my mind is that (I've read this in my textbook, but I have to prove it) if A is infinite, then it admits a bijection with a proper subset; but I'm not sure if proving that would help me to automatically say that 1) implies 2) and 3) because A\{x} and A\{x1,...,xn} are proper subsets of A, but how do I know that these are the indicated proper subsets, I mean, the statement says that if a set is infinite, it admits a bijection with A proper subset, not every proper subset.
Now, when it comes to prove that 2) or 3) imply 1) I am totally stuck. So, that's all that I have, if I think of something, I'll post it here.

 

[verty]

Do the 1 -> 2 proof so long, you seem to have solid ideas about that. When you've done that, you'll likely know more and will find a proof for 2 -> 1.

PS. I say this but the Dedekind-infinite -> infinite direction will be tough, probably you'll need to look it up.

 

[Mandelbroth]

Homework Statement

Let A be a set, prove that the following statements are equivalent:

1) A is infinite
2) For every x in A, there exists a bijective function f from A to A\{x}.
3) For every {x1,...,xn} in A, there exists a bijective function from A to A\{x1,...xn}

Relevant equations

The first and only thing that comes to my mind is that (I've read this in my textbook, but I have to prove it) if A is infinite, then it admits a bijection with a proper subset; but I'm not sure if proving that would help me to automatically say that 1) implies 2) and 3) because A\{x} and A\{x1,...,xn} are proper subsets of A, but how do I know that these are the indicated proper subsets, I mean, the statement says that if a set is infinite, it admits a bijection with A proper subset, not every proper subset.
Now, when it comes to prove that 2) or 3) imply 1) I am totally stuck. So, that's all that I have, if I think of something, I'll post it here.

These are not necessarily equivalent. For example, let $A=\emptyset$. If $A$ is empty, the bijection in 2 and 3 is just the empty function given by $f:\emptyset\to\emptyset$.

 

[mahler1]

These are not necessarily equivalent. For example, let $A=\emptyset$. If $A$ is empty, the bijection in 2 and 3 is just the empty function given by $f:\emptyset\to\emptyset$.

Sorry, I forgot I've posted this. Now I am trying to solve the problem again. I think that in the case A is the empty set, 2 and 3 don't make any sense, because what is A-{x} or A-{x1,...,xn} if the empty set doesn't have any elements which belong to it?

 

[Mandelbroth]

Sorry, I forgot I've posted this. Now I am trying to solve the problem again. I think that in the case A is the empty set, 2 and 3 don't make any sense, because what is A-{x} or A-{x1,...,xn} if the empty set doesn't have any elements which belong to it?

It's still the empty set. It's what we call a "vacuous truth." The exact statements say "for every $x$ in $A$..."

If there are no elements in $A$, then the statement must be true for "all" the elements in $A$ because there aren't any elements to satisfy it.

Even if we just have the statements without the "for every $x$ in $A$" requirement, the empty function is trivially a bijection.