Mathematical representation

Mathematical statement

The same mathematical object can be represented in words, symbols, tables, ordered pairs or diagrams. To determine whether representations correspond, first fix the object, domain, scale and interpretation rules; visual resemblance alone does not establish equivalence.

Representative formula (exact TeX)

R(f)

Assumptions

• Exercises use integer-coefficient affine rules and complete explicitly supplied finite tables or ordered pairs.
• Audit coordinate and table data only on the listed domain. Unlisted points and graph appearance supply no implicit values.

Symbol scope

• R — representation record
• f — represented mathematical object

Intuition, definition, and misconceptions

A table lists specified inputs and outputs, whereas a symbolic expression gives a computation rule. They form an explicit semantic bridge only on a common domain with agreement in every row.

Two points determine a nonvertical affine rule only after checking that their x-coordinates differ. Repeated inputs with different outputs do not form a function table.

A diagram can aid understanding, but pixel locations, apparent straightness and unmarked scales cannot replace explicit data.

Do not infer formula equivalence merely from visually similar graphs or overlook domains and conflicting repeated inputs.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Variables and algebraic expressions — Symbolic representations use expressions with variables.
• Function mapping — A function table must assign exactly one output to each input.
• Coordinate plane — Ordered pairs and coordinates use ordered components.

Verification boundary

Agreement in a finite table covers only the explicit domain; it implies neither unlisted values nor educational outcomes.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Develop more complete typed semantic bridges among diagrams, language, symbols and data tables.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.