Problem 24 - 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 \(c .\) $$ ^{\prime}\left\\{^{\prime}\right. $$

Step-by-Step Solution

Answer

The function values for each character are: \(g(',') = 44\), \(g('{') = 123\), and \(g('}') = 125\).

Step 1: Determine ASCII values of the given characters.

We can consult an ASCII table to find the ASCII values of the characters ',', '{', and '}'. Using an ASCII table, we can find the following ASCII values: - The ASCII value of ',' is 44. - The ASCII value of '{' is 123. - The ASCII value of '}' is 125.

Step 2: Calculate the function values g(c) for each character.

According to the exercise, the function g(c) maps the given characters to their ordinal numbers in the ASCII table. This means that the function g(c) gives us the ASCII values of the characters, which we have already determined in Step 1. Thus, we have: - \(g(',') = 44\) - \(g('{') = 123\) - \(g('}') = 125\) Now, we have determined the value of the function g(c) for each character in the given set.

Key Concepts

Ordinal Numbers

Ordinal numbers are fascinating because they indicate a position or order of elements in a set. In the context of characters and encoding, an ordinal number represents the position of a character within the ASCII table. This sequence helps computers understand characters systematically. For instance, if we consider the alphabet, 'A' is the first letter, which can be represented by the ordinal number 1 if we're mapping its position. However, in ASCII terms, 'A' corresponds to the value 65 because ASCII starts with non-visible characters before reaching the visible ones. Understanding ordinal numbers in encoding is crucial because it allows programmers and computers to process text information consistently.

Character Mapping

Character mapping is an essential concept in computer science, allowing the conversion between characters and their numeric representations. With character mapping, each character is assigned a unique number, which can be referred to as its ASCII value. This is akin to giving each character in a language its distinctive GPS coordinates in the digital world. This process simplifies how systems input, output, or analyze text data. For instance, the character 'A' maps to the number 65 in the ASCII table. Similarly, the comma ',' is mapped to 44, making it easier for machines to manage text." Essentially, character mapping links human-readable characters with computer-operable numeric codes, bridging the gap between human expression and machine processing.

ASCII Table

The ASCII table is a standardized chart utilized in computing to represent characters numerically. Let's delve into its structure: It includes 128 unique codes, from 0 to 127, encompassing letters, digits, punctuation, and various control characters. Here's how the ASCII table works:

• Start of Table (0-31): These are control characters, used for text formatting and communication in devices.
• Printable Characters (32-126): This range includes printable symbols such as letters (both upper and lower case), numerals, and punctuation marks.
• Last Value (127): Known as the delete control character.

The ASCII table is essential for text data transmission between different systems as it provides a universal standard. Without such a guide, sharing information between diverse devices and programs would be chaotic. By representing characters in a numerical form, ASCII aids in storing and transmitting text in a consistent manner across computing platforms.