Definition:Induction Step
Terminology of Mathematical Induction
Consider a proof by mathematical induction:
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 $w_0$.
The proof that the truth of $Q$ for $w_p$ implies the truth of $Q$ for $w_{p^+}$is called the induction step.
Expressed in the various contexts of mathematical induction:
First Principle of Finite Induction
The step which shows that $n \in S \implies n + 1 \in S$ is called the induction step.
First Principle of Mathematical Induction
The step which shows that $\map P k \implies \map P {k + 1}$ is called the induction step.
Second Principle of Finite Induction
The step which shows that $n + 1 \in S$ follows from the assumption that $k \in S$ for all values of $k$ between $n_0$ and $n$ is called the induction step.
Second Principle of Mathematical Induction
The step which shows that the truth of $\map P {k + 1}$ follows from the assumption of truth of $P$ for all values of $j$ between $n_0$ and $k$ is called the induction step.
Principle of General Induction
The step which shows that $\map P x = \T \implies \map P {\map g x} = \T$ is called the induction step.
Principle of General Induction for Minimally Closed Class
The step which shows that $\map P x = \T \implies \map P {\map g x} = \T$ is called the induction step.
Principle of Superinduction
The step which shows that $\map P x = \T \implies \map P {\map g x} = \T$ is called the induction step.
Also known as
The induction step can also be referred to as the inductive step.
Also see
Sources
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction