MHB What is the meaning of one-to-one correspondence between subsets of S?

[yakin]

What is the meaning of one-to-one correspondence between subsets of S?

 

[Ackbach]

So, a function you can think of as a subset of points in the $xy$ plane that satisfy a vertical line test: no vertical line you draw can touch the function in more than one place. If, in addition, the function passes a horizontal line test (no horizontal line you could draw touches the function in more than one place), then we say the function is one-to-one. When you say "one-to-one correspondence", you typically mean a one-to-one function.

More formally: a one-to-one function is a function that maps unique points in the domain to unique points in the range. (And of course, if it's a function, then unique points in the range had to come from unique points in the domain.)

 

[Evgeny.Makarov]

When you say "one-to-one correspondence", you typically mean a one-to-one function.

Wikipedia says that one-to-one correspondence is a bijective function, i.e., a function that is both one-to-one and onto. I've seen this interpretation before as well. And yes, it is confusing.

 

[Ackbach]

Wikipedia says that one-to-one correspondence is a bijective function, i.e., a function that is both one-to-one and onto. I've seen this interpretation before as well. And yes, it is confusing.

Sure, although in this context, it likely means a function between two infinite sets that allows you to conclude that they have the same cardinality. Is "onto" a necessary property to show this?

 

[Evgeny.Makarov]

Of course: the fact that two sets have the same cardinality is evidenced by a bijection, and by definition it is both one-to-one and onto.

In any case, this is the terminology that I've seen so far.

• One-to-one function: a function that maps different inputs to different outputs.
• One-to-one correspondence: a function that is one-to-one and onto, i.e., covers the whole codomain.