Definition:Basis for the Induction
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 step that establishes the truth of $Q$ for $w_0$ is called the basis for the induction.
Expressed in the various contexts of mathematical induction:
First Principle of Finite Induction
The step that shows that the integer $n_0$ is an element of $S$ is called the basis for the induction.
First Principle of Mathematical Induction
The step that shows that the proposition $\map P {n_0}$ is true for the first value $n_0$ is called the basis for the induction.
Second Principle of Finite Induction
The step that shows that the integer $n_0$ is an element of $S$ is called the basis for the induction.
Second Principle of Mathematical Induction
The step that shows that the proposition $\map P {n_0}$ is true for the first value $n_0$ is called the basis for the induction.
Principle of General Induction
The step that shows that the propositional function $P$ holds for $\O$ is called the basis for the induction.
Principle of General Induction for Minimally Closed Class
The step that shows that the propositional function $P$ holds for the distinguished $b \in M$ is called the basis for the induction.
Principle of Superinduction
The step that shows that the propositional function $P$ holds for $\O$ is called the basis for the induction.
Also known as
The basis for the induction is often informally referred to as the base case.
Also see
Sources
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction