The fundamental number sets

0 := βˆ… = {} Zero 1 = {βˆ…} One 𝔹 = {βŠ₯, ⊀}

β„• = {1,2,3,...} β„€ = {...,-1,0,1,...} β„š = { p/q | p,q ∈ β„€, q β‰  0 } ℝ β„‚

ℕᐩ = β„• {0} β„•β‚€ = β„• ⋃ {0} β„• = ℀ᐩ β„™ = { p ∈ β„• | βˆ€x ∈ β„•. x βˆ‰ {1,p} -> x ∀ p } = primes ℀⁻ = β„€ β„•β‚€

π•Ž = β„• ⋃ {0} ℕᐩ = β„• {0} β„€ = ℀⁻ ⋃ {0} ⋃ ℕᐩ

β„š = β„€ ⋃ 𝔽 β„š = fromList [ p % q | p <- [1..], q <- [1..] ] :: Set Integer

ℝ = β„š ⋃ 𝕀 ℝ = 𝕋 ⋃ 𝔸 ℝ = ℝ⁻ ⋃ {0} ⋃ ℝᐩ β„‚ = ℝ ⋃ 𝕁

  • subset-of relations are intuitively: β„• βŠ† β„€ βŠ† β„š βŠ† ℝ βŠ† β„‚

  • however, unintuitively: β„• = β„€ = β„š < ℝ < β„‚

ℕᐩ ℕ⁻ β„•οΉ’ β„•Λ— ℀ᐩ ℀⁻ β„€οΉ’ β„€Λ— β„šα© β„šβ» β„šοΉ’ β„šΛ— ℝᐩ ℝ⁻ ℝ﹒ ℝ˗ ℂᐩ ℂ⁻ β„‚οΉ’ β„‚Λ—

𝔸 𝔹 β„‚ 𝔻 𝔼 𝔽 𝔾 ℍ 𝕀 𝕁 𝕂 𝕃 𝕄 β„• 𝕆 β„™ β„š ℝ π•Š 𝕋 π•Œ 𝕍 π•Ž 𝕏 𝕐 ℀​

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