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Contents Index Dark Mode Prev Up Next \(\newcommand{\im}{{\mathrm{im}}}
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\)
Chapter 3 Induction
Many times, a universal quantification can be reinterpreted as a nice family of propositions, indexed by positive integers. For example, consider the statement “the sum of the first \(n\) positive integers is \(\frac{n(n + 1)}{2}\text{.}\) ” This can be rephrased as the universally quantified statement “for every positive integer \(n\text{,}\) \(1 + 2 + \dots + n=\frac{n(n + 1)}{2}\text{.}\) ” This can be seen as a family of infinitely many simpler propositions:
\begin{align*}
1 \amp = 1 \\
1 + 2 \amp = \frac{2(2 + 1)}{2} \\
1 + 2 + 3 \amp = \frac{3(3 + 1)}{2} \\
\amp \vdots
\end{align*}
Mathematical induction is a powerful proof technique that allows us to prove certain universally quantified statements by examining closely the related family of more basic propositions. Despite its name, mathematical induction is an example of deductive, not inductive, reasoning.
Outcomes
Reproduce the principle of mathematical induction
Prove theorems using mathematical induction.
