Problem 22 - Discrete Mathematics with Applications

Question

Let \(g: \mathrm{ASCII} \rightarrow A\) defined by \(g(c)=n,\) where \(A=\\{32,33, \ldots, 126\\}\) and \(n\) denotes the ordinal number of the character \(c .\) Find \(g(c)\) for each character \(c .\) $$'<'$$

Step-by-Step Solution

Answer

To find the value of \(g(c)\) for each character in the ASCII character set (from 32 to 126), simply determine the ordinal number of the character, which is its position in the ASCII character set. For example, \(g('A') = 65\), \(g('B') = 66\), and \(g('C') = 67\).

Step 1: Understand the Function domain

The function \(g(c)\) is defined for any character \(c\) in the ASCII character set. The domain of this function is the ASCII character set, which ranges from 32 to 126.

Step 2: Finding the ordinal number of a character

The ordinal number of a character can be found by simply determining its position in the ASCII character set. For example, the ordinal number of 'A' is \(65\), as 'A' holds the 65th position in the ASCII character set.

Step 3: Computing g(c) for each character

Now that we understand how to find the ordinal number of each character, we can compute the value of \(g(c)\) for each character. For example, let's compute the values for the following characters: 1. \(c = 'A' \): The ordinal number of 'A' is 65, thus \(g(A) = 65\). 2. \(c = 'B' \): The ordinal number of 'B' is 66, thus \(g(B) = 66\). 3. \(c = 'C' \): The ordinal number of 'C' is 67, thus \(g(C) = 67\). Similarly, we can compute the value of \(g(c)\) for each character in the ASCII character set.

Key Concepts

Ordinal Number

An ordinal number in the context of ASCII refers to the position of a character within the ASCII character set. This specific number uniquely identifies each character based on their order. Think of it like a seat number in a theater. For example, in the ASCII table:

• The number 65 refers to the letter 'A'
• The number 66 refers to the letter 'B'
• The number 67 corresponds to 'C'

Each character is assigned such a number, making it possible to reference them in binary code representations. Understanding these numbers is crucial because they form the basis of many computations that involve character manipulation.

Character Mapping

Character mapping is the process of linking characters to their respective ordinal numbers in a set. In ASCII, each character is assigned a unique number. This mapping ensures a consistent way for digital systems to interpret and store text data. For example:

• 'A' is mapped to 65
• 'B' is mapped to 66
• 'C' is mapped to 67

This mapping is essential for software applications that require text handling, allowing computers to efficiently process and output human-readable text by referencing these ordinal mappings.

Function Domain

In mathematics and programming, the domain of a function is the set of all possible inputs that the function can accept. For the function \(g(c)\) related to ASCII, the domain consists of all valid ASCII characters ranging from 32 to 126. This means that the function can accept any character within this range. Here’s what it means practically:

• The function won't process characters outside this range
• It ensures all characters are within the standard displayable ASCII set

The function domain here provides a boundary, ensuring that inputs are valid thus preventing errors that might arise from processing undefined characters.

ASCII Character Set

The ASCII character set is a standardized collection of characters used in computers and digital systems. It stands for American Standard Code for Information Interchange. The set includes various characters such as uppercase and lowercase English letters, digits, punctuation, and control characters. ASCII originally had a range from 0 to 127, but the exercise focuses on the printable characters 32 to 126. This set includes:

• Letters like 'A' to 'Z' and 'a' to 'z'
• Numbers from '0' to '9'
• Symbols like !, @, #, etc.

The importance of ASCII lies in its widespread adoption as a basic textual interchange format, enabling different systems to communicate text without compatibility issues.