Definition:Mathematical Induction

Proof Technique

Mathematical induction is a proof technique which works in two steps as follows:

$(1): \quad$ A statement $Q$ is established as being true for some distinguished element $w_0$ of a well-ordered set $W$.

$(2): \quad$ A proof is generated demonstrating that if $Q$ is true for an arbitrary element $w_p$ of $W$, then it is also true for its immediate successor $w_{p^+}$.

The conclusion is drawn that $Q$ is true for all elements of $W$ which are successors of $w_0$.

It is established in a number of contexts, according to how it is to be used.

Principle of Finite Induction

Let $n_0 \in \Z$ be given.

Let $\Z_{\ge n_0}$ denote the set:

$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$

Let $S \subseteq \Z_{\ge n_0}$ be a subset of $\Z_{\ge n_0}$.

Suppose that:

$(1): \quad n_0 \in S$

$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$

Then:

$\forall n \ge n_0: n \in S$

That is:

$S = \Z_{\ge n_0}$

Principle of General Induction

Let $M$ be a class.

Let $g: M \to M$ be a mapping on $M$.

Let $M$ be minimally inductive under $g$.

Let $P: M \to \set {\T, \F}$ be a propositional function on $M$.

Suppose that:

\((1)\)	$:$					\(\ds \map P \O \)	\(\ds = \)	\(\ds \T \)
\((2)\)	$:$			\(\ds \forall x \in M:\)		\(\ds \map P x \)	\(\ds = \)	\(\ds \T \implies \map P {\map g x} = \T \)

Then:

$\forall x \in M: \map P x = \T$

Principle of Mathematical Induction

Let $\map P n$ be a propositional function depending on $n \in \Z$.

Let $n_0 \in \Z$ be given.

Suppose that:

$(1): \quad \map P {n_0}$ is true

$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$

Then:

$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$.

Second Principle of Finite Induction

Let $S \subseteq \Z$ be a subset of the integers.

Let $n_0 \in \Z$ be given.

Suppose that:

$(1): \quad n_0 \in S$

$(2): \quad \forall n \ge n_0: \paren {\forall k: n_0 \le k \le n \implies k \in S} \implies n + 1 \in S$

Then:

$\forall n \ge n_0: n \in S$

Second Principle of Mathematical Induction

Let $\map P n$ be a propositional function depending on $n \in \Z$.

Let $n_0 \in \Z$ be given.

Suppose that:

$(1): \quad \map P {n_0}$ is true

$(2): \quad \forall k \in \Z: k \ge n_0: \map P {n_0} \land \map P {n_0 + 1} \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$

Then:

$\map P n$ is true for all $n \ge n_0$.

This process is called proof by (mathematical) induction.

Terminology

Basis for the Induction

The step that establishes the truth of $Q$ for $w_0$ is called the basis for the induction.

Induction Hypothesis

The assumption that $Q$ is true for $w_p$ is called the induction hypothesis.

Induction Step

The proof that the truth of $Q$ for $w_p$ implies the truth of $Q$ for $w_{p^+}$is called the induction step.

Also see

• Do not confuse with Definition:Inductive Argument.

• Results about Proofs by Induction can be found here.