Regarding cardinality and mapping between sets.

[Terrell]

why is not always true that if $\vert A\vert\leq\vert B\vert$ then there exist an injection from $A$ to $B$?

 

[member 587159]

why is not always true that if $\vert A\vert\leq\vert B\vert$ then there exist an injection from $A$ to $B$?

Who told you that?

By definition $|A| \leq |B|$ iff there exists an injection $A \to B$

 

[Terrell]

Who told you that?

By definition $|A| \leq |B|$ iff there exists an injection $A \to B$

I think saying that me saying that it's not always true is too strong of a statement, but what really happened is I couldn't find any sources that mentions that it must be an if and only if statement. Thank you a lot! It has cause me a lot of unnecessary thinking lol!

 

[member 587159]

I think saying that me saying that it's not always true is too strong of a statement, but what really happened is I couldn't find any sources that mentions that it must be an if and only if statement. Thank you a lot! It has cause me a lot of unnecessary thinking lol!

That's common in mathematics. When one writes a definition one often says "if" while it should be "iff". I remember it confused me in the beginning too!