Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 4 / 5E / b / ii / Solution

Again let $A=\{a_1,\dots ,a_m\}$ with $m>1$, and enumerate $B=\{b_1,b_2,\dots \}$. For each infinite binary sequence $x=(x_j{)}_{j\ge 1}$, define a function $g_x$ byThe first $m$ values make every $g_x$ a surjective function, while the remaining values recover $x$ uniquely. Hence $x\mapsto g_x$ is an injective function from the uncountable set of binary sequences into $S_2$. This proves $S_2$ is an uncountable set, an instance of uncountability of surjections onto a nontrivial finite set.