#quality varies

[cashmerecorey]

INBOW TRICK OR TREATING

come into my inbox and say trick or treat— i’ll do a DC doodle (and i truly mean doodle, do not expect quality) of something i think you’d like based on your blog :) they will not be finished on halloween day cus i’m busy but YEAH! Ask me!!

[anremithrl]

Math notation

Recently saw a shitpost with funny math notation sent to me by @another-s1lly-being, and while it was funny, I was disappointed it wasn't an actual math notation post. To share the joy, I've mocked up a quick math notation post. I've tried to separate it by branch of mathematics and which theory or language it comes from. Feel free to add with notation from various fields. This is not meant super rigorously or to be a replacement for any classes, it's more of a glossary. This is also not meant to be in depth at all. It is aimed primarily at non-mathematicians.

I will format each entry like so:

Name

Symbol Use Read as

Meaning

[Additional Note]

(This went through several iterations because I could not get it to fit in my phone's screen.)

A hodge-podge glossary of math notation.

Mathematics mostly studies abstraction and precise ideas. At early levels, the first ideas made precise are the notions of abstract quantities.

For example, the number 3, is the abstraction used to discuss anything that there might be 3 of. 3 units of length. 3 bricks. Etc. Then we learn addition, which is the abstraction and formalisation of the idea of putting things together. Etc.

Since mathematics deals a lot with finding out things about truth and whether things are true, we have a bunch of notatuon to help us quickly communicate ideas about proofs.

Basic Proofs:

These are the notations used to talk about truth, and uses to denote derrivations of truths, "proofs". The notations are a lot more in depth than just what I wrote here.

Tautology

⟙ ⟙ true

Represents a true statement.

Contradiction

⟘ ⟘ false

Represents a statement that is not true.

Semantic Entailment

⊨ Γ⊨Δ entails

At least one statement in Δ is true in every model where every statement in Γ is true.

Syntactic Entailment

⊢ Γ⊢Δ proves

You can prove one of the statements of Δ using the statements of Γ.

(Logical) Negation

￢ ￢P not

This is true if and only if P is not true.

Standard ￢ Truth Table.

P ￢P

⟙ ⟘

⟘ ⟙

(Logical) Disjunction

⋁ P⋁Q or

This is true exactly when at least one of P or Q is true. Notably this includes the case where they are both true.

Standard ⋁ Truth Table

P Q P⋁Q

⟙ ⟙ ⟙

⟙ ⟘ ⟙

⟘ ⟙ ⟙

⟘ ⟘ ⟘

(Logical) Conjunction

⋀ P⋀Q and

This one is true exactly when both of them are true. You can remember it is "and" because it is in the same direction as an "A".

Standard ⋀ Truth Table

P Q P⋀Q

⟙ ⟙ ⟙

⟙ ⟘ ⟘

⟘ ⟘ ⟘

⟘ ⟘ ⟘

Existential Quantifier

∃ ∃x Φ(x) [there] exists-such that

This is true if and only if there exists some object that satisfies the condition Φ.

For example ∃x x²=2 is a statement that is true if and only if there exists some x such that x²=2.

"There exists some object which we will label as x such that that object squared equals two."

Universal Quantifier

∀ ∀x Φ(x) for all

Thia one is true if the condition given is always satisfied. For example:

For example, ∀x x=x, "for all x, x is equal to x", is true no matter what you chose for x.

Conditional

⇒ P⇒Q implies/if-then

This means that Q must be true if P is. Notably, Q being true does not mean anything about P. When P us false, this is trivially true. Can be written backwards for reverse meaning. P⇐Q is the same as Q⇒P. Can be read as P because Q, or P is implied by Q.

P⇒Q "if P then Q"

Standard ⇒ Truth Table

P Q P⇒Q

⟙ ⟙ ⟙

⟙ ⟘ ⟘

⟘ ⟙ ⟙

⟘ ⟘ ⟙

Biconditional

⇔ P⇔Q if and only if

This means that P is true exactly when Q is true. P and Q have the same truth value.

Standard ⇔ Truth Table

P Q P⇔Q

⟙ ⟙ ⟙

⟙ ⟘ ⟘

⟘ ⟙ ⟘

⟘ ⟘ ⟙

Equality

= a=b equals

This one is true exactly when a and b are the same object.

(Modal) Necessity

□ □P necessarily

This means that P is necessarily true in all worlds.

(Modal) Possibility

◇ ◇P possibly

This means that the statement P is possible in some world.

Note, there are other variants of a lot of this notation. And other unmentioned notation.

Set Theory:

We like talking about collections of objects as well. We cannot easily talk about having 3 things without being able to have three things. We call these collections 'sets'. We can explicitly write down sets this way: {0,1,2}. Sets are not allowed to contain themselves.

Explicit Set Notation

{} {x₁, x₂, ...} set of

Directly building a set with elements x₁, x₂, etc.

Membership

∈ x∈y in, element of, member of

This is the same as saying x is in the set y.

For example, if we look at the collection of even numbers (2ℤ), 4∈2ℤ.

Another example 1∈{1,2}

Sidenote:

At this point I also feel like we have developed enough vocabulary and notation to express a more complex idea in this language.

Let us express the idea we call extensionality, which is something we want to be true about sets. Extensionality means any two sets are the same if they have all the same elements. In other words, order or arrangement do not matter to sets, they only know what is in them.

∀x∀y(∀z z∈x⇔z∈y)⇒x=y

For all x and y, if for every z, z is in x if and only if z is in y, then x is equal to y.

Color breakdown.

∀x∀y(∀z z∈x⇔z∈y)⇒x=y

For all x and y, if for every z, z is in x if and only if z is in y, then x is equal to y. (literal translation)

For any two sets, if any object in either one is also in the other, then they are the same set. (english translation)

These notations gives you a shorthand to talk about complex ideas. It is the language we use to talk about math.

∈ on its own is enough to describe everything about sets, but it is easier to use shorthands.

Back on track

Subset

⊂ x⊂y subset

x is a subset of y is a shorthand that means that any element in x is also contained in y. Can be written backwards for reverse meaning, called superset.

In other words, x⊂y is shorthand for ∀z z∈x ⇒ z∈y. This is different from equality because it's only a conditional, not a biconditional.

For example {1,2}⊂{1,2,3} or even {1,2}⊂{1,2}

Proper Subset

⊊ x⊊y proper subset

This means that x is a subset of y, but is not equal to y.

x⊊y is shorthand for x⊂y⋀x≠y.

Powerset

𝒫 𝒫(x) subsets of, powerset of

The set of all the subsets of x.

That can be described compactly with this rule:

∀z z∈𝒫(x)⇔z⊂x

(Set) Disjunction

∪ x∪y, ∪x union

The union of x and y is the set all the elements that appear in any of them. When used on a single element, it refers to the union of all of its internal elements.

∀z (z∈x∪y) ⇔ (z∈x⋁z∈y)

∀z (z∈∪x)⇔(∃y z∈y⋀y∈x)

(Set) Conjunction

∩ x∩y, ∩x intersect

The intersection of x and y contains all the elements that are in both of them. When used on just x, gives the set of all elements which are inside every element of x.

∀z (z∈x∩y)⇔(z∈x⋀z∈y)

∀z (z∈∩x)⇔(∀y y∈x⇒z∈y)

Separation

⎮ y⎮Φ filtered by, such that

The set of elements in y that fulfill condition Φ.

∀z (z∈y⎮Φ)⇔(z∈y ⋀ Φ(y))

Replacement

: f:x image

The set of results of f applied to every element in x.

∀z (z∈f:x)⇔(∃y y∈x⋀f(y)=z)

These can be combined, or done anonymously, via set-builder notation.

{f(x):x∈y⎮Φ(x)} is the same thing as f:(x⎮Φ)

Setminus

\ x\y without, removed

This is the set of elements in x without the elements in y. If y⊂x, we call this the complement of y in x.

∀z (z∈x\y)⇔(z∈x⋀z∉y)

Or, using easier notation.

x\y={z∈x⎮y∉x}

Symmetric Difference

⨆ x⨆y difference

The elementa that are in either, but not both. The elements in exactly one.

x⨆y=(x∪y)\(x∩y)

(Set) Cross Product

× x×y cross

The set of all ordered pairs of elements. Choosing one element from x and one from y.

x×y={(a,b):a∈x,b∈y}

For example, the set of points on a 2-d plane can be called ℝ×ℝ.

(Set) Function

→ x→y to

The set of all functions from x to y.

Empty Set

⦰ ⦰ empty

The set that contains nothing.

∀z z∉⦰

Successor

⁺ x⁺ successor

This is the set containing both x and the elements of x. This comes up a lot because it conveniently always makes a different set, with exactly one more element.

x⁺=x∪{x}

This one is super important. If you have a set x with n elements, x⁺ has n+1 elements. This gives you the canonical representation of the natural numbers and with union and axiom of infinity gives the ordinals.

A Few Relevant Named Sets:

Symbol Name Set

Example construction

ℕ Natural Numbers {0, 1, 2, 3...}

Let Φ(x)≔⦰∈x ⋀ (∀y y∈x⇒y⁺∈x)

By axiom of infinity there must be some ω' that Φ(ω').

ω=∩(𝒫(y)⎮Φ)

Note that Φ(ω) and Φ(x)⇒ ω⊂x are very good exercises to prove.

ω is a set that can represent ℕ because they both include 0 and then count up. Via ⦰~0, n⁺∽n+1, and are minimal.

ℤ Integers {...-3,-2,-1,0,1,2,3...}

ℤ⋍2×(ℕ\1)∪1

Theze are Z via positive, negative, and 0.

ℚ Rationals {p/q:p∈ℤ,q∈ℤ⎮q≠0}

While my favourite method is finite sequences, the easiest:

ℚ'=ℤ×(ℤ\1)

ℚ⋍{{(a',b')∈ℚ'⎮a'b=ab'}:(a,b)∈ℚ'}

ℝ Reals (-∞,∞)

Φ(x)≔∀y,z∈ℚ ((y∈x ⋀z＜y)⇒(z∈x)) note you do have to define less than for ℚ.

𝒫(ℚ)⎮Φ

[rychuart]

Assorted splatoon sketchies, mostly pearlina because they live in my skull

[patchofvoid]

An array of art I did for the first arc of our Ravenloft campaign

[botwstoriesandsuch]

hows the ask box progress going?

its not my faULt that five of you sent me great asks that warrant my essay level responses okAy

[bonos-grindcore-sideproject]

Bono, Popmart, Mexico City

[favouriteculturalpastimeib]

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