One to One Correspondence vs One to One function

[bigpunz04]

What is the difference between the two?

The topic we are currently reading about is Set Cardinality. There are a couple of definitions listed in the book that seem to define them as different properties of sets. Is there a difference between the two or are they different terms with the same meaning? See below:

Definition 1
The sets A and B have the same cardinality if and only if there is a "one-to-one correspondence" from A to B. When A and B have the same cardinality, we write |A| = |B|

Definition 2
If there is a "one-to-one function" from A to B, the cardinality of A is less than or the same as the cardinality of B and we write |A| <= |B|. Moreover, when |A|<=|B| and A and B have different cardinality, we say that the cardinality of A is less than the cardinality of B and we write |A|<|B|

Thank you!

 

[Evgeny.Makarov]

One-to-one function is otherwise called an injection. One-to-one correspondence is called a bijection. It is an injection that is also a surjection. I agree that the English names are somewhat confusing.