Notation and Symbols

Operations

Wikipedia: Mathematical operators and symbols in Unicode Wikipedia: Supplemental Mathematical Operators

⊕ - XOR. Direct Sum.
⊖ - Symmetric difference.
⊗ - Tensor product.
⊘ -
⊙ - Circled dot operator. ⨀ n-ary circled dot operator.
∗ - Astrisk Operator. Also ✱.
∘ - Function composition. Ring operator. (Also ∙ bullet operator)
⋈ - Natural join.
⋆ - Star Operator.

Calculus

∫ ∬ ∭ ∮ ∯ ∰ - Integrals (double+triple). ∮ ∯ ∰ - Contour, surface and volume integrals. ∱ ∲ ∳ - Clockwise integral, clowise contour integral. [1]
∆ - ∆x change in x. Sometimes symmetric difference.
∇ - Gradient.

Other

≠ ≈ ≡ ∃ ∄ ∈ ∉ ∌ ⇔ ∀ ∧ ∨ ¬ ≤ ≥ ≮ ≯ ⇒ ∩ ∪ ⋂ ⋃ ⊃ ⊂ ⊇ ⊆ ⊢

Arithmetic: − + ± × ⋅ · ÷ ⁄ √ ∛ ∜
x′ - x prime.

Misc: ∑ ℵ ℶ ∞ ∴ ∵ ! ∎ → ⟨ ⟩ ㏒ * ∥ ∦

𝓋⃗, v⃗ - Vector 𝓋 + (U+20D7).
∅ - empty set
∂ - Partial Derivative.
δ - Kronecker Delta
∏ - Product over (Like ∑)
⊤ - True
⊥ - False
∖ - Set minus.
σ - Selection
{♠, ♦, ♥, ♣}

Subscript: ₀₁₂₃₄₅₆₇₈₉ ₊₋₌₍₎ ₐₑₒₓₔₕₖₗₘₙₚₛₜ
Superscript: ⁰¹²³⁴⁵⁶⁷⁸⁹ ⁺⁻⁼⁽⁾ ᵃᵇᶜᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖʳˢᵗᵘᵛʷˣʸᶻ
Blackboard Bold: 𝔸𝔹ℂ𝔻𝔼𝔽𝔾ℍ𝕀𝕁𝕂𝕃𝕄ℕ𝕆ℙℚℝ𝕊𝕋𝕌𝕍𝕎𝕏𝕐ℤ 𝕒𝕓𝕔𝕕𝕖𝕗𝕘𝕙𝕚𝕛𝕜𝕝𝕞𝕟𝕠𝕡𝕢𝕣𝕤𝕥𝕦𝕧𝕨𝕩𝕪𝕫 𝟘𝟙𝟚𝟛𝟜𝟝𝟞𝟟𝟠𝟡 ℾℽℿℼ⅀ ⅅⅆⅇⅈⅉ

wikipedia
Mathematical Italic: 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻𝐼𝐽𝐾𝐿𝑀𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈𝑉𝑊𝑋𝑌𝑍𝑎𝑏𝑐𝑑𝑒𝑓𝑔ℎ𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝𝑞𝑟𝑠𝑡𝑢𝑣𝑤𝑥𝑦𝑧

|𝐴|×|𝐵|

Number Systems

ℕ - The set of all natural numbers. No zero. No negatives. Subtraction and division aren't always possible. ℕ = {1, 2, 3, ...}.
ℤ - The set of all integers. Subtraction is always possible. ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}. ℤ is an infinite ring.
ℤₘ - The set of integers modulo m. A finite ring.
ℚ - The set of all rational numbers (Quotient). Fractions. Division is always possible. ℚ is a field and an infinite ring.
ℝ - The set of all real numbers. Includes irrational and transcendental numbers. √(2), π, e, φ. ℝ is an extension of ℚ to a larger field. Also an infinite ring.
ℝ⁺ - The set of all positive reals.
ℂ - The set of all complex numbers. i²=-1. Is a field and an infinite ring.
ℍ - The set of all Quaternions.
𝔽ₚ - Set of integers modulo a prime p. Addition and multiplication also defined modulo p. A field.

ℚ(γ) - γ is the root of the polynomial x³-x-1. When γ³ appears it is replaced with γ+1. See: 2.2 in princeton companion of mathematics. AUTOMORPHISM
ℙ - Primes.

Algebraic Structures

Properties

associative - a(bc) = (ab)c
commutative - ab = ba

homomorphism - A category of function. Preserves the structure. f: X→Y. f(a)f(b)=f(c) and ab=c. A homomorphism of vector spaces is refered to as a linear map.
bijection - A bijection is invertable.
isomorphism - A homomorphism that is also a bijection. An operation-preserving bijection. A homomorphism with an inverse that is also a homomorphism. f: X→Y, g: Y→X
automorphism - Galios Groups related. An isomorphism from a structure to itself.
Abelion - An abelian group/operation is commutative.

Structures

Set

A collection of elements.

Group

An algebraic structure that is a set with a single binary operation.

An infinite group has an unlimited number of elements. (ℤ, +)

A finite groups has a finite number of elements. (ℤₘ, +) - The integers mod m.

The order of an element is the number of elements in the subgroup it generates. |2| = 3

(𝔾, ∗) - Group 𝔾 with operation ∗.

Axioms
- The operation must be closed. - a,b ∈ 𝔾 → a∗b ∈ 𝔾
- The operation must be associative. - (a∗b)∗c = a∗(b∗c)
- Operation must have an identity element that has no effect on any other element under operation. ∃e(a∗e = e∗a = a).
- Every element must have an inverse. An element when combined with the original will return the identity. ∀a∃a⁻¹(a∗a⁻¹=a⁻¹∗a=e)

(ℤ, +) - The integers under addition.
- Closure: a,b ∈ ℤ → a+b ∈ ℤ
- Associativity: (a+b)+c = a+(b+c)
- Identity: a+0 = 0+a = a
- Inverse: a⁻¹ = -a

(ℤ, ×) - The integers under multiplication it not a group
- Closure: a,b ∈ ℤ → a×b ∈ ℤ
- Associativity: (a×b)×c = a×(b×c)
- Identity: a×1 = 1×a = a
- NO Inverse: 2x=1 has no solution in ℤ

If the operation is also commutative then it is an Abelian group.
Bill Shillito.

Ring

Similar to a field but multiplication doesn't require an inverse.

ℤ, ℚ, ℝ, ℂ, ℤₘ are all unital commutative rings.

𝕄₂(ℝ) the set of all 2×2 real matrices is a non-commutative ring as matrix multiplication is not commutative.

+ must be abelian.
× must be closed and associative.
Bill Shillito - Rings and Fields

(R, +, *)
- (R, +) is an abelian group - addition
  - Closed
  - Associative
  - Identity - Additive identity
  - Inverse - Additive inverse −a
  - Commutative
- (R, *)
  - Closed under ×
  - Operation × must be associative.
- + and × must be linked by the distributive property. Multiplication distributes over addition.
  - a×(b+c) = a×b+a×c
  - (a+b)×c = a×c+b×c
    - 0×a = a×0 = 0 - Zero times anything is zero
    - a×−b = −a×b = −(a×b) - A positive times a negative is a negative.
    - −a×−b = a×b - A negative times a negative is a positive.

× might not be commutative. If it is then it is a commutative ring (not called abelian which is only for groups).
If the ring has a multiplicative identity then it is a unital ring. For ℤ, ℚ, ℝ, ℂ, ℤₘ the multiplicative identity is 1. For 𝕄ₘ(ℝ) it is the identity matrix.
An element of a unital ring that has a multiplicative inverse is called a unit. Not every element is necessarily a unit. In ℤ only 1 and −1.
If every element other than 0 is a unit. It is a division ring. ℚ is a division ring.
A zero divisor of a ring is a nonzero element that can be multiplied by some other nonzero element to produce 0. In ℤ₆, 2, 3 and 4 are zero divisors. 2×3=0
An element can not be a unit and a zero divisor.
An integral domain is a commutative, unital ring with no zero divisors. ℤ, ℚ, ℝ, ℂ are integral domains.
- ℤₘ is not necessarily an integral domain. ℤₚ is an integral domain if p is a prime number.
- In an integral domain you can cancel factors from both sides of an equation. 2×x=2×3

Polynomial Ring

A monomial is the product of a number and a non-negative integer power of a variable. eg: 2x³

A polynomial is a finite sum of monomials. a₀+a₁x+a₂x²+a₃x³+…+aₙxⁿ

n is the degree of the polynomial. It's highest power.

Some polynomial rings: ℤ[x], ℚ[x], ℝ[x], ℂ[x], ℤₘ[x]

Field

Like a ring with stricter multiplication axioms.

A field is a division ring where × is commutative, unital and has no zero divisors and every nonzero element is a unit.

A field forms an abelian group under both addition and multiplication.

ℚ, ℝ, ℂ are fields.

ℤₚ is a field if p is prime.

ℤ is not a field.

A set with 2 binary operations (+, ×)
Both must be commutative and associative
And have an identity elements (+ is 0 and × is 1)
Every element must have an inverse (x = -x, x = 1/x)
Must follow the distributive law
Must be closed under addition, multiplication, taking of inverses (Results must in in the same set).

Vector Space

- Like a 2D or 3D plane. Can be built from 'unit vectors' or 'basis'. for example (1,0), (0,1).
- There was a Youtube on the topic (Blue2Brown1).
- Scalar - A number used to multiply a vector. Infinite dimensional vector spaces exist such as when vectors are functions.
- ℝⁿ - A vector space of n-dimensions over the field ℝ. ℝ is the scalar type. ℝ², ℝ³, etc...

Groups

Dihedral Groups: Dₙ - D₃ is the dihedral group of the triangle.
Symmetry Groups: Sₙ - S₃ the symmetric group of 3 elements. S₃ is isomorphic to D₃. (S₃ ≅ D₃)
Special Orthogonal: SO(n) - special orthogonal group (rotation group?).
General Linear: GLₙ(ℝ) - The group of all n×n matrices with non-zero determinants under matrix multiplication. nonabelian group because matrix operations are non-commutative.

Cayley table

A table showing all the results of all possible operations on a finite group.

Eignvectors

Eigenvector - A vector in a vector space doesn't get rotated by a linear transformation. It stays on it's 'span'. It's the axis of rotation.
Eigenvalue - The value an Eigenvector is scaled by. 𝛢𝓋⃗=λ𝓋⃗. 𝛢 is transformation matrix. 𝓋⃗ is an Eigenvector. λ is the eigenvalue.