---
type: math
---
## Induction
Similar (if not the same) to:
- [Mathematical Proofs (Induction)](Mathematical%20Proofs%20(Induction).md)
- [Structural Proofs](Proofs.md)


- Base case $0\in \mathbb{N}$
- Inductive step - if $n\in \mathbb{N} \implies n+1\in \mathbb{N}$
- We allow a finite number of steps


I.e.
Given $f (n) = n(n + 1)$ for all $n\in N$, then $f (n)$ is even.

**Base case:** $f(0) = 0\times 1 = 0$, which is even
**I.S.:**
$$
f(n+1) = (n+1)(n+2)= n(n+1)+2(n+1) = f(n) + 2(n+1) \blacksquare
$$

## Strings and Languages
Literally the same as [Mathematical Data Structures](Mathematical%20Data%20Structures.md), but on strings

How to define the reversal of a string, inductively?


Let $w$ be a finite string. We define $w^R$ by induction on $|w|$:

**B.C.:**
$|w| = 0$, then, trivially, $w = \epsilon \therefore w^R = \epsilon$

**I.S.:**
$|w| = n \geq 1$, so $w = u a$ with $|u| = n-1$,


## Operations on strings
- Concatenation (associative)
- Substring, prefix, suffix
- Replication (exponentiation): a string concatenated with itself
- Reversal ($u^R$)

## Operations on languages
- Lifting operations on strings to languages
- Concatenation of languages
- Kleene star - $V^*$ -> smallest superset[^1] of V that contains the empty string and is closed under concatenation, i.e. one or more repetitions
- Reversal

## Regular sets / languages
This used to be in DS, but I don't have it in this repo.

Recursively defined over an alphabet $\Sigma$ from 
- $\emptyset$
- $\{\epsilon\}$
- $\{a\} | \forall a \in \Sigma$

Regex is  a notatio nto denote regular languages, i.e.:

[^1]: The opposite of a subset - a set which contains all elements of another (and possibly more)