# Definition:Principle of Mathematical Induction/Induction Step

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## Terminology of Principle of Mathematical Induction

Consider a **Proof by Mathematical Induction**.

The step which shows that $\map P k \implies \map P {k + 1}$ is called the **induction step**.

## Also presented as

Some sources present the **induction step** in the form:

- $\map P {k - 1} \implies \map P k$

## Also known as

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

## Sources

- 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.1$. Arithmetic - 1979: John E. Hopcroft and Jeffrey D. Ullman:
*Introduction to Automata Theory, Languages, and Computation*... (previous) ... (next): Chapter $1$: Preliminaries: $1.3$ Inductive Proofs - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.4.2$: Mathematical induction: Definition $1.29$