Let $S$ Be the Set of All Strings in 0's and 1's

question 7

Essay

Let $S$ be the set of all strings in 0's and 1's, and define a function $F : S \rightarrow \mathbf { Z } ^ { \text {nonneg } }$ as follows: for all strings $s$ in $S$ ,
$$F ( s ) = \text { the number of } 1 \text { 's in } s .$$
(a) What is $F ( 001000 ) ? F ( 111001 )$ ? $F ( 10101 )$ ? $F ( 0100 )$ ?
(b) Is $F$ one-to-one? Prove or give a counterexample.
(c) Is $F$ onto? Prove or give a counterexample.
(d) Is $F$ a one-to-one correspondence? If so, find $F ^ { - 1 }$ .

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