Surjection between two finite sets of different sizes Suppose I have two sets $A$ and $B$. $A = \{1, 2\}$ and $B = \{2, 4, 6, 8\}$. I am trying to prove that there is no surjection from $A$ to $B$ by finding at least one $y$ value from $B$ such that for every element $x\in A$,  $f(x)\ne y$. I am not entirely sure how to prove this and kind of stuck.

A fully rigorous argument here can be extremely hard.  To give a mostly rigorous argument:  Suppose for contradiction that $f:A\to B$ is a surjection.  Then $f(A)⊆ B$ and $f(A)=\{f(1),f(2)\}$ so that there are only two elements in $f(A)$.  Since there are four elements in $B$ then there must be some element in $B\smallsetminus f(A)$ (the "relative complement").  If we call this element $y$ then there is no $x$ such that $f(x)=y$.
