Consider the function f that maps strings over $\{$a$,$ b$,$ c$\}$ to strings over $\{0,1\}$ by replacing $4$pts each a by $0,$ each b by $0,$ and each c by $1.$ For example f$($cabbc$)=10001.$ The function f extends naturally to languages: if A is a language over $\{$a$,$ b$,$ c$\},$ then f$($A$)$ is the language over $\{0,1\}$ defined as $\{$f$($w$)|$ w is in A$\}.$ Prove: If a language A is recognizable, then so is f$($A$).$