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 $F ^ { - 1 }$ .

---

Definitions:

Achievement Motivation

The desire to accomplish something significant, excel, and reach high standards of performance.

Fear of Failure

The pervasive apprehension about failing to meet standards or achieve goals, which can significantly inhibit one's willingness to take risks or try new tasks.

Botany Course

An educational class dedicated to the scientific study of plant life, including their classification, structure, processes, and contributions to the environment.

Subjective Conscious Experience

An individual's personal, internal experience of phenomena, which cannot be directly observed or measured by others.