Napkin notes - https://usamo.files.wordpress.com/2016/07/napkin-2016-10-02.pdf

gistfile1.txt

# Sets

ℕ = set of positive integers
ℤ = set of all integers
ℂ = set of all complex numbers
ℝ = set of all real numbers
ℚ = set of all rational numbers
∅ = {}
H = { x | ℕ > 1 } == H is the set of all postive integers greater than 1 (| == 'such that')
x ∈ ℕ == x belongs to ℕ
A ⊆ B == A is a subset of B (all elements of A occur in B)
A ⊊ B == A is a proper subset of B (all elements of A occur in B, but A != B)
A ∪ B == union
A ∩ B == intersection
A \ B == in A but not B

# Functions

f(A) = B == mapping of every element of set A to set B. 
f : A → B == A is the domain of f, B is the codomain.
f(a) = b or a ↦ b says that f takes a to b
injective: when a1 != a2 then f(a1) != f(a2). Or, f: A ↪ B (every a maps to a different b, i.e. a 1:1 mapping)
surjective: b ∈ B, a ∈ A and f(a) = b. Or, f: A ↠ B (you can map every a to a b. more than 1 a can map to the same b)
bijection: for every b ∈ B there is a unique a ∈ A such that f(a) = b (every a is mapped to a unique b)

e.g. bijection: f(a) = a + 5 for A = {1,2,3,4,5}, B = {6,7,8,9,10}
e.g. surjective but not injective: f(a) = a*a where a ∈ ℤ and b ∈ ℕ

* If f: A ↪ B is injective, then the size of A is at most the size of B
* If f: A ↦ B is surjective, then the saize of A is at least the size of B
* If f: A → B is a bijection, then the size of A is the size of B
* If f: A → B is a bijection, then there is a inverse function which maps f: B → A