Formal Proof: Roman Letters to ASCII
A rigorous demonstration of the mapping between alphabetic characters and their ASCII representations
Introduction
This document provides a formal proof of the mapping between the first nine Roman letters (a-i) and their corresponding ASCII symbol representations. ASCII (American Standard Code for Information Interchange) is a character encoding standard that assigns unique numeric codes to characters.
Definitions
L = {a, b, c, d, e, f, g, h, i} — The set of first nine lowercase Roman letters
A — The set of ASCII codes representing characters
f: L → A — The mapping function from letters to ASCII codes
Formal Proof
Theorem
There exists a bijective function f: L → A that maps each Roman letter in L to a unique ASCII code in A.
Proof
1. By definition, ASCII assigns unique numeric codes to each character.
2. Let f(l) be defined as the function that returns the ASCII code of letter l.
3. To prove injectivity: Assume f(l₁) = f(l₂). Since ASCII codes are unique, l₁ = l₂.
4. To prove surjectivity: For each ASCII code in the range, there exists a corresponding letter in L.
5. Therefore, f is bijective. ∎
Mapping Table
The following table demonstrates the mapping f: L → A for each letter in L:
| Roman Letter | ASCII Symbol | ASCII Code (Decimal) | ASCII Code (Binary) | HTML Entity |
|---|---|---|---|---|
| a | a | 97 | 01100001 | a |
| b | b | 98 | 01100010 | b |
| c | c | 99 | 01100011 | c |
| d | d | 100 | 01100100 | d |
| e | e | 101 | 01100101 | e |
| f | f | 102 | 01100110 | f |
| g | g | 103 | 01100111 | g |
| h | h | 104 | 01101000 | h |
| i | i | 105 | 01101001 | i |
Mathematical Representation
Formal Definition of the Mapping
Let L = {l | l is a lowercase Roman letter from a to i}
Let A = {n | n ∈ ℕ, 97 ≤ n ≤ 105}
The function f: L → A is defined as:
f(l) = 96 + ord(l) - ord('a') + 1, where ord(l) returns the position of l in the alphabet.
More simply: f(l) = 96 + pos(l), where pos(l) is the position of the letter in the alphabet (a=1, b=2, ..., i=9).
Proof of Correctness
For l = 'a': pos('a') = 1, so f('a') = 96 + 1 = 97 ✓
For l = 'b': pos('b') = 2, so f('b') = 96 + 2 = 98 ✓
For l = 'c': pos('c') = 3, so f('c') = 96 + 3 = 99 ✓
...
For l = 'i': pos('i') = 9, so f('i') = 96 + 9 = 105 ✓
Properties of the Mapping
Bijectivity
The function f is both injective and surjective:
Injective (one-to-one): No two different letters map to the same ASCII code.
Surjective (onto): Every ASCII code from 97 to 105 has a corresponding letter.
Therefore, f is bijective, and an inverse function f⁻¹: A → L exists.
Order Preservation
The function f preserves the natural ordering of letters:
If l₁ comes before l₂ in the alphabet, then f(l₁) < f(l₂).
This makes f an order-isomorphism between L and A.
Conclusion
We have formally proven the existence of a bijective mapping between the set of the first nine Roman letters L = {a, b, c, d, e, f, g, h, i} and their corresponding ASCII codes A = {97, 98, 99, 100, 101, 102, 103, 104, 105}.
The mapping function f(l) = 96 + pos(l) is both injective and surjective, preserving the natural ordering of the letters. This provides a rigorous foundation for the digital representation of textual information using the ASCII standard.
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