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beckyconning/minimal-maximal.txt

Created September 30, 2017 02:42
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minimal-maximal.txt
let P = {{1,2,3}, {2,3} {2,4}}
m is a maximal element of if ∀ s. s ∈ S ⇒ m ≤ s implies m = s
(P, ≤):
S¹ S² S³
S¹ 1 0 0 🤷 ∀x(x ∈ {(S¹, S¹)} → fst x = snd x) = {(S¹, S¹)} ∴ S¹ is a maximal element of P
S² 1 1 0 🤷 ∀x(x ∈ {(S², S¹), (S², S²)} → fst x = snd x) ≠ {(S², S¹), (S², S²)} ∴ S¹ is NOT a maximal element of P
S³ 0 0 1 🤷 ∀x(x ∈ {(S¹, S¹)} → fst x = snd x) = {(S¹, S¹)} ∴ S¹ is a maximal element of P
m is a minimal element of if ∀ s. s ∈ S ⇒ m ≥ s implies m = s
(P, ≥):
S¹ S² S³
S¹ 1 1 0 🤷 {(S¹, S¹), (S¹, S²)}
S² 0 1 0 🤷 ∀x(x ∈ {(S², S²)} → fst x = snd x) = {(S², S²)} ∴ S² is a minimal element of P
S³ 0 0 1 🤷 ∀x(x ∈ {(S³, S³)} → fst x = snd x) = {(S³, S³)} ∴ S³ is a minimal element of P
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