# Definition:Induction Hypothesis

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

Expressed in the various contexts of mathematical induction:

### First Principle of Finite Induction

The assumption made that $n \in S$ for some $n \in \Z$ is the **induction hypothesis**.

### First Principle of Mathematical Induction

The assumption made that $\map P k$ is true for some $k \in \Z$ is the **induction hypothesis**.

### Second Principle of Finite Induction

The assumption that $\forall k: n_0 \le k \le n: k \in S$ for some $n \in \Z$ is the **induction hypothesis**.

### Second Principle of Mathematical Induction

The assumption that $\forall j: n_0 \le j \le k: \map P j$ is true for some $k \in \Z$ is the **induction hypothesis**.

### Principle of General Induction

The assumption made that $\map P x$ is true for some $x \in M$ is called the **induction hypothesis**.

### Principle of General Induction for Minimally Closed Class

The assumption made that $\map P x$ is true for some $x \in M$ is called the **induction hypothesis**.

### Principle of Superinduction

The assumption made that $\map P x$ is true for some $x \in M$ is called the **induction hypothesis**.

## Also known as

The **induction hypothesis** can also be referred to as the **inductive hypothesis**.

## Also see

## Sources

- 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction