Strings and languages:
- An alphabet is a finite set of symbols, such as {1,2,3} or {0,1}
- A string is an alphabet $\sum$ is a finite sequence of symblols from $\sum$, for example 010010 and 10010101 are strings over the alphabet {0,1}
- A languge is a set of strings
	- L1 = {00, 01, 10, 11}
	- L2 = {a^n | n is an integer non-negative}
Notation: a to the power of n just means a repeated n times

Examples
$L_8 = \{a^m b^n | m,n \in \mathbb{z}^{nonneg}\}$

So valid strings would be
aaabbb, aaabbbb, ab, abb

$L_9 = \{a^n b^n | n \in \mathbb{z}^{nonneg}\}$

So valid strings would be
aabb, ab, aaaabbbb, aaaaaabbbbbb


$L_{11} = \{w \in \{a, b\}^* | \text{The number of "a"s is equal to the number of "b"s}\}$
This would include everything that L9 includes, but allows for the alphabet can be out of order

You can also written this as
$L_{11} = \{w \in \{a, b\}^* | n_a(w) = n_b(w)\}$


You can extend the repetition notation (exponent) to include multiple symbols by wrapping it in parentheses

Example:
$L_{12} = \{(ab)^n | n \in \mathbb{z}^{nonneg}\}$

This means we need to be careful as 
$L_{13} = \{ab^n | n \in \mathbb{z}^{nonneg}\}$
is different, as L12 is asking for repetition of ab, while L13 is assigning a prefix of a and repetition of b 

$L_{14} = \{a^i b^j c^k | i,j,k \in \mathbb{z}^{nonneg}, j = 2i + 3k\}$

So this means that for any number of "a"s and "c"s, the number of "b"s is predetermined if you have selected your "a"s and "c"s

Intro to functions and relations

Informally, a function (f) descrives an input output situation (A -> B)
For every input element a in A, there is exactly one ouput element b in B

f maps a to b

A is the domain of f
B is the co-domain of f