Problem 17 - Discrete Mathematics with Applications

Question

Let \(A=\\{32,33, \ldots, 126\\} .\) Let \(f: A \rightarrow\) ASCII defined by \(f(n)=\) character with ordinal number \(n .\) Find \(f(n)\) for each value of \(n\). $$38$$

Step-by-Step Solution

Answer

The ASCII character associated with the ordinal number 38 is the '&' symbol. Therefore, \(f(38) = '&'\).

Step 1: Identifying the given ordinal number

In this exercise, we are given the ordinal number n as 38. We are required to find the ASCII character associated with this ordinal number.

Step 2: Look up the ASCII table

To find the character associated with the ordinal number 38, we will consult the ASCII table. There are many online resources which provide ASCII tables, or you can refer to a physical ASCII table if you have one available.

Step 3: Find the character for the given ordinal number

After consulting the ASCII table, we can see that the character associated with the ordinal number 38 is the '&' symbol.

Step 4: Write down the result

After completing the above steps, we can now conclude that for n=38, the function f(n) maps to the '&' symbol. In mathematical notation, we can write this as: \[f(38) = '&'\]

Key Concepts

Discrete Mathematics

Discrete mathematics is a branch of mathematics that deals with objects that can assume only distinct, separated values. It encompasses a wide range of topics, including logic, set theory, graph theory, and number theory, among others.

Within this field, functions play a pivotal role, especially when dealing with finite sets. In the exercise, we observe a function mapping from a set of integers to ASCII characters. This relates to discrete math in that the set of ASCII characters is finite and each character can be distinctly identified by its unique ordinal number, fitting well within the discrete framework.

Functions

A function in mathematics is a relation between two sets that assigns to each element of the first set, called the domain, exactly one element of the second set, known as the codomain. In simpler terms, it's like a machine that takes an input, does some specific operation, and gives out a corresponding output.

In our given exercise, the function is denoted by f, which takes an ordinal number n from set A as input and outputs an ASCII character. This shows how functions can be modeled to represent all sorts of relationships, including those between numbers and symbols.

Ordinal Numbers

Ordinal numbers are used to describe the order of objects in a sequence, such as first, second, third, and so on. In the context of computer science and mathematics, ordinals are usually represented as integers, where they can also serve as indices representing positions or sequence.

In character encoding systems like ASCII, each symbol is assigned a unique ordinal number to differentiate it from others. The exercise deals with the ordinal number 38, which is used to locate the corresponding ASCII character.

Character Encoding

Character encoding is a system that pairs each symbol in a character set with a unique number, allowing digital devices to represent and manipulate text. ASCII, which stands for American Standard Code for Information Interchange, is one of the earliest character encodings and is limited to 128 unique symbols.

ASCII includes letters, digits, punctuation marks, and control codes, each mapped to a number from 0 to 127. The exercise requires the knowledge of ASCII encoding to find what character is represented by the ordinal number 38, where each number aligns with a specific character.