mathematical Induction (algebra) assume there is a function like $f:A→B$ which is injective, why it means $\left|A\right|\le\left|B\right|$ or in another way why a function like $g:B→A$ 
 stands for $\left|B\right|\le\left|A\right|$
my problem has been clearly explained here: 
assume a set like $A=[1,2,3,4]$ and $B=[1,2,3]$ and for $f:A→B$ we have $f=[(1,1)(2,2),(3,3)]$ and $f$ is still injective but $\left|A\right|=4$ and $\left|B\right|=3$ and in this case $\left|B\right|\le\left|A\right|$...

In old books you can find a difference between function and aplication.

  
*
  
*A function $f\colon A\to B$ has as domain a subset of $A$. 
  
*An application $f\colon A \to B$ has as domain all $A$

for people, in general , those means the same.
If you want to speak strictly, the intuition behind injectivity is: 

You can find a "copy" of the domain of $f\colon A \to B$ in the set $B$

