Under which conditions do injections f and g imply the existence of an injection h : B → C?

Question

Under which conditions do injections   f and   g imply the existence of an injection   h  :  B  →  C?

Paola Mercer · Accepted Answer

A  ,  B  ,  C  are sets.There exists an injection   f  :  A  →  B.There exists an injection   f  :  A  →  B.There exists an injection   g  :  A  →  Cthen all that says is that       |    A      |    ≤      |    B      |   and       |    A      |    ≤      |    C      |  .For any sets   B  ,  C, either       |    B      |    ≤      |    C      |   or       |    C      |    ≤      |    B      |   (or both), but with the given information, you can&#039;t decide which one, hence you don&#039;t know whether ot not there is an injection   h  :  B  →  CHowever, if it&#039;s given that f is not only injective but also surjective (and hence bijective), then       |    A      |    =      |    B      |  . Then since       |    A      |    ≤      |    C      |  , we get       |    B      |    ≤      |    C      |  , hence there is an injection   h  :  B  →  C.Along the same lines, if it&#039;s given that   f and   g are both bijective, then       |    A      |    =      |    B      |   and       |    A      |    =      |    C      |  , hence       |    B      |    =      |    C      |  , so there is a bijection from   B to   C, which implies the existence of injections   h  ,  k which you asked for.

Under which conditions do injections f and g imply the existence of an injection h:B rarr C?

Open question

Answer & Explanation

New Questions in High school geometry