37 lines
907 B
Markdown
37 lines
907 B
Markdown
---
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type: math
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---
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## Induction
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Similar (if not the same) to:
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- [Mathematical Proofs (Induction)](Mathematical%20Proofs%20(Induction).md)
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- [Structural Proofs](Proofs.md)
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- Base case $0\in \mathbb{N}$
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- Inductive step - if $n\in \mathbb{N} \implies n+1\in \mathbb{N}$
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- We allow a finite number of steps
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I.e.
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Given $f (n) = n(n + 1)$ for all $n\in N$, then $f (n)$ is even.
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**Base case:** $f(0) = 0\times 1 = 0$, which is even
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**I.S.:**
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$$
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f(n+1) = (n+1)(n+2)= n(n+1)+2(n+1) = f(n) + 2(n+1) \blacksquare
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$$
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## Strings and Languages
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Literally the same as [Mathematical Data Structures](Mathematical%20Data%20Structures.md), but on strings
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How to define the reversal of a string, inductively?
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Let $w$ be a finite string. We define $w^R$ by induction on $|w|$:
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**B.C.:**
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$|w| = 0$, then, trivially, $w = \epsilon \therefore w^R = \epsilon$
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**I.S.:**
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$|w| = n \geq 1$, so $w = u a$ with $|u| = n-1$,
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