Let S be the set of all strings in a $\text{'}$s and b $\text{'}$s$,$ and define C: S $->$ S by C$($s$)=$a s$,$ for each s in s$.($ C is called concatenation by a on the left.$)($a$)$ Is C one$-$to$-$one? To answer this question, suppose s$\_1$ and s$\_2$ are strings in s such that C$($s$\_1)=$C$($s$\_2).$ Use the definition of C to write this equation in terms of a$,$ s$\_1,$ and s$\_2$ as follows. a s$\_1=$ each integer n $>=0,$ the nth character from the left in s$\_1$ the nth character from the s$\_2.$ Hence, s$\_1?$ s$\_2,$ and so C one$-$to$-$one. $($b$)$ Show that C is not onto. Counterexample: The string is in S but is not equal to C$($s$)$ for $-$ Select$-.$ string s because $-$ Select$-$ with