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

question 18

Essay

Let $S$ be the set of all strings in 0 's and 1's, and define a function $g : S \longrightarrow \mathbf { Z }$ as follows: for each string $s$ in $S$ ,
$$g ( s ) = \text { the number of } 0 \text { 's in } s .$$
(a) What is $g ( 101011 )$ ? $g ( 00100 )$ ?
(b) Is $g$ one-to-one? Prove or give a counterexample.
(c) Is $g$ onto? Prove or give a counterexample.

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Definitions:

Financial Leverage

The use of borrowed funds to increase the potential return of an investment, amplifying both potential gains and losses.

Net Income

The total earnings of a company after subtracting all expenses from revenue, including taxes and interest.

Debt Financing

A method of raising capital through the sale of bonds, bills, or notes to individual and/or institutional investors, in return for lending the company money.

Operating Leverage

A measure of how revenue growth translates into growth in operating income, indicating the proportion of fixed versus variable costs a company has.