What properties does $A\to B$ satisfy under 1-1 correspondence?

Question

A 1-1 correspondence between two sets $A$ and $B$ is a function $f\colon A \to B$ satisfying what properties?

I do know that we say that two sets $A$ and $B$ are equivalent, and we write $A \sim B$ iff they can be put into 1-1 correspondence. This is really an equivalence relation: $A \sim A$

If $A\sim B$ then $B \sim A$ $\sim$ is symmetric.
If $A\sim B$ and $B\sim C$ then $A\sim C$ $\sim$ is transitive

I guess I am having trouble interpreting what the the initial question is asking?

I am trying to do this so I can prove that $|A| \leq |A|$.

Thank you.

Answer 1

A one to one correspondence indicates that there is a map $f: A\to B$ that is onto, so that for every $b\in B$ there is an $a\in A$ such that $f(a) = b$, and also one to one, meaning that the $a\in A$ such that $f(a) =b$ is unique.

Answer 2

If you want to show that $A\sim A$, all you have to do is find a function from $A$ to itself which is both injective and surjective.

For example, the identity function defined by $f(a)=a$.

Answer 3

A 1-1 correspondence is another word for bijective. Thus, it is one-one and onto. A one-one function is a mapping such that f(x1) = f(x2) if and only if x1 = x2. An onto mapping is a mapping such that every element is mapped over. So suppose we have f = x + 1 and our set is [1,2,3,4]. Then our new set would be [2,3,4,5] so we see that this is a 1-1 corresondence.