approx 5

• 01 August 2025
• Mathematics,
• Process book,
• Learning

Today I am playing ⛵️Alcumus🐬, to up my problem solving skills!

I believe it is cognitively restorative, and #maybe useful for succeeding at BSc exams.

-- with steel drums

another fascinating idea: - prealgebra is #maybe the theory semirings/monoids?

structure Monoid := (M : Type) (op : M → M → M) (id : M) (assoc : ∀ x y z, op x (op y z) = op (op x y) z) (id_l : ∀ x, op id x = x)

🥏

What is the chance of heads over tails when flipping nothing?

"Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that the penny and dime both come up the same? "

https://artofproblemsolving.com/alcumus/problem

*There is maybe A x B or A + B here, product, coproduct, does probability theory have topos logic? *

events = subobjects (ℰ ⊆ Ω)

probabilities = measures = morphisms

μ : Ω → [0,1] ⊆ ℝ

--- Wow, that's like [H,T]! 🦎

[H,T] ≅ Bool ≅ subobject classifier Ω

--- Wooooooooow! A coin is a `subobject classifier Ω ! 🪙

🪙coin ~ Ω

🤖🌿 H ↔ true, T ↔ false → flipping = sampling truth!
   coins classify subobjects in probabilistic worlds

Is ex falso quodlibet same as vacuously true?

• ⊥ ⊢ φ for any φ - logic explosion, from falsehood anything follows

• ∀x∈∅, P(x) ⇒ ✅ true, ∅ has no counterexample

  falsehood vs. emptiness ≠ same

Nice 🎉: #maybe flipping 4 coins:
  f : 1 ⟶ Ω⁴

π₁, π₃ : Ω⁴ → Ω --extract penny & dime.

equal : Ω × Ω ⟶ Ω  
equal(h, t) := h ↔ t

P(penny = dime) := μ ∘ equal ∘ (π₁ × π₃) ∘ f

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