# Proofs

Proofs are a concept in discrete mathematics used to verify the truthfulness and validity of statements. They use logical reasoning to establish the certainty of mathematical claims, playing a crucial role in advancing the field of mathematics.

## Conjecture

Conjectures are used to construct a mathematical proof. They can be thought of as a building block of a hypothesis that can be validated using logic and patterns.

Mathematical exploration and discovery heavily depend on the implementation of conjecture.

## Base Case

Base case is term that is described as the initial or smallest value that verifies a statement to be correct. According to the Principle of Mathematical Induction, a proof is bifurcated into two steps:

• Base Case (verification of base step): The initial or smallest value is checked to establish the truth of the statement.
• Inductive Step (inductive hypothesis and conclusion): Assuming the statement holds for a given value, it is shown that it also holds for the next value, extending the validity of the statement.

## Proofs

Proof by Strong Induction
An induction technique that proves a statement by providing multiple base cases, assuming the statement is true for all integers from the largest base case to some even larger integer k, and then proving the statement is true for k+1 using that assumption.