Let S be the set of all strings in a$\text{'}$s and b$\text{'}$s$,$ and define

C: S $->$ S

by

C$($s$)=$ as$,$

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.

as$1=$

Correct: Your answer is correct.

Now strings are finite sequences of characters, and since the strings on both sides of the above equation are equal, for each integer

n $>=0,$

the nth character from the left in the left$-$hand string

equals

Changed: Your submitted answer was incorrect. Your current answer has not been submitted.

the nth character from the

left

Correct: Your answer is correct.

in the right$-$hand string. It follows that for each integer

n $>=0,$

the nth character from the left in

s$1$

equals

Changed: Your submitted answer was incorrect. Your current answer has not been submitted.

the nth character from the

left

Correct: Your answer is correct.

in

s$2.$

Hence,

s$1$

$=$

Correct: Your answer is correct.

s$2,$

and so C

is

Correct: Your answer is correct.

one$-$to$-$one.

$($b$)$

Show that C is not onto.

Counterexample: The string

Incorrect: Your answer is incorrect.

is in S but is not equal to

C$($s$)$

for

any

Correct: Your answer is correct.

string s because

every string in the range of C starts

Correct: Your answer is correct.

with

Incorrect: Your answer is incorrect.

$.$