MHB One to One Correspondence vs One to One function

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One-to-one correspondence and one-to-one function are distinct concepts in set theory. A one-to-one correspondence (or bijection) indicates that two sets have the same cardinality, meaning there is a perfect pairing between their elements. In contrast, a one-to-one function (or injection) implies that the cardinality of one set is less than or equal to that of another, without guaranteeing a perfect pairing. The terminology can be confusing, as one-to-one functions do not necessarily imply equal cardinality, while one-to-one correspondence does. Understanding these definitions is crucial for discussing set cardinality accurately.
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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!
 
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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.
 
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