MONAD GRAPhi-X
Vedecké laboratórium · jeden graf na stránku
Logika a množiny
Science lab page · one study surface at a time.
Sets · living Venn
A ∪ B, A ∩ B, A △ B, |A × B| = |A|·|B|
|A| = 6 |B| = 5 |A∩B| = 1 |U| = 12
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{ n ∈ U : 2 | n }
{ n ∈ U : n is prime }
A
{ 2, 4, 6, 8, 10, 12 } |A| = 6
B
{ 2, 3, 5, 7, 11 } |B| = 5
A ∪ B
{ 2, 3, 4, 5, 6, 7, 8, 10, 11, 12 }
A ∩ B
{ 2 }
A \\ B
{ 4, 6, 8, 10, 12 }
A △ B
{ 3, 4, 5, 6, 7, 8, 10, 11, 12 }
Aᶜ
{ 1, 3, 5, 7, 9, 11 }
|P(A)|
2^6 = 64
Cartesian product A × B (first 24 ordered pairs, |A × B| = 30)
(2,2) (2,3) (2,5) (2,7) (2,11) (4,2) (4,3) (4,5) (4,7) (4,11) (6,2) (6,3) (6,5) (6,7) (6,11) (8,2) (8,3) (8,5) (8,7) (8,11) (10,2) (10,3) (10,5) (10,7) …
Semantics: A ⊆ B is false, B ⊆ A is false. Two sets are equal exactly when each is a subset of the other (extensionality). De Morgan: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ — verified on this universe of 12 elements.