The Binomial theorem - Vocabulary

Vocabulary on Induction proofs

• Base Case: The initial step of an induction proof, demonstrating that the statement holds for the first value.
• Conclusion: The final part of the induction proof, restating that the statement holds for all intended values (initially defined by the exercise).
• Conjecture: A mathematical statement which appears to be true, but has not been formally proven. A conjecture can be thought of as the mathematicians way of saying “I believe that this is true, but I have no proof yet”. A conjecture is a good guess or an idea about a pattern.
• Hypothesis (Inductive hypothesis): The assumption made during the inductive step, assuming the statement is true for some arbitrary case.
• Induction Principle: The logical principle allowing conclusions about infinite sets by establishing a base case and an inductive step.
• Inductive Step: The part of an induction proof where the truth of the statement for the next case is demonstrated based on the inductive hypothesis.
• Inequality: A mathematical relation used frequently in induction to demonstrate the validity of a statement through comparison.
• Integers: Represented by $\mathbb{Z}$, the set of integers includes all positive and negative whole numbers, including zero. Mathematically, they are defined as $\displaystyle \mathbb{Z} = \left\{\ldots,-1, 0, 1, 2, \ldots \right\}$.
• Natural Numbers: The most typical set over which induction proofs are performed, denoted by $\mathbb{N}$. The set of natural numbers (according to the IB standard) are $\mathbb{N} = \left\{ 0, 1, 2, \ldots \right\}$ (however typically the set of natural numbers does not include zero).
• Proof by Mathematical Induction: A structured method for proving a mathematical statement holds true for all integers greater than or equal to a certain initial integer.
• Proposition: A statement that is either true or false. For example “Portugal is in Europe” is a true statement and “All primes are odd numbers” is a false statement. We use $P(n)$ ("P of n") to denote a propositional function (a formula which becomes true or false if you plug in a value for $n$).
• Recursive Relation: A formula or relation in induction that defines each term based on preceding terms.