Definitions, Theorems, and Proofs

In theoretical computer science, definitions, theorems, and proofs form the core framework for describing and verifying mathematical concepts.

1. Fundamental Components

Definitions: Precise descriptions of objects and notions. They must be unambiguous, clearly stating what constitutes an object and what does not.
Theorems: Precise mathematical statements that have been proven true. They typically express that a certain object possesses a specific property.
Lemmas: Mathematical statements that are primarily interesting because they assist in proving a more significant theorem.
Corollaries: Statements that can be easily concluded as true directly from a theorem or its proof.

A proof is a convincing logical argument demonstrating that a statement is absolutely true.

Standard of Rigor: Unlike legal or everyday proofs, which may rely on “beyond reasonable doubt” or a “weight of evidence,” a mathematical proof must be airtight and convincing in an absolute sense.
Role of Evidence: In mathematics, empirical evidence plays no role; an argument is only accepted if it is logically certain.

Finding a proof is often a creative process of discovery rather than a linear task. A general strategy for approaching a proof:

Understand the Statement: Ensure you fully grasp the definitions of all terms involved.
Experiment: Try small examples to see why the statement might be true (or if you can find a counterexample).
Break it Down: Divide the problem into smaller, manageable parts.
Be Patient: Proof construction often involves trial and error; if one approach fails, try looking at the problem from a different angle.

Term	Brief Description
Set	A group of objects.
Function	An operation translating inputs into outputs.
Graph	A collection of nodes and edges connecting them.
Boolean Value	TRUE or FALSE (often represented as 1 or 0).
Alphabet	A finite, nonempty set of symbols.
Language	A set of strings.