Definition:Mathematical Induction

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

Let $n_0 \in \Z$ be given.


Suppose that:

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


Then:

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


That is:

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


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

  • Results about Proofs by Induction can be found here.