1.7 KiB
type
| type |
|---|
| math |
Induction
Similar (if not the same) to:
-
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, 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 superset1 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.:
-
The opposite of a subset - a set which contains all elements of another (and possibly more) ↩︎