13397593
Beschreibung
Mindmap von Luke Byrne, aktualisiert more than 1 year ago
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Erstellt von Luke Byrne
vor fast 7 Jahre
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Example Proof in Set Theory
- Proposition: "∀A, B sets, (A ∩ B) ∪ (A\B) = A".
- Show (A ∩ B) ∪ (A\B) ⊆ A
- ∀x ∈ (A ∩ B) ∪
(A\B), x ∈ (A ∩ B)
or x ∈ A\B
- If x ∈ (A∩B), then clearly x ∈ A as A∩B ⊆ by definition.
- If x ∈ A\B, then by definition, x ∈ A and x !∈ B, so definitely x ∈ A.
- In both cases, x ∈ A as needed.
- In both cases, x ∈ A as needed.
- If x ∈ A\B, then by definition, x ∈ A and x !∈ B, so definitely x ∈ A.
- If x ∈ (A∩B), then clearly x ∈ A as A∩B ⊆ by definition.
- ∀x ∈ (A ∩ B) ∪
(A\B), x ∈ (A ∩ B)
or x ∈ A\B
- Show A ⊆ (A ∩ B) ∪ (A\B)
- Either...
- x ∈ B
- ... then x ∈ A and x ∈ B, so x ∈ A ∩ B
- ... in both cases
- x ∈ (A ∩ B) or x ∩ (A\B)
- so x ∈ (A ∩ B) ∪ (A\B), as needed
- so x ∈ (A ∩ B) ∪ (A\B), as needed
- x ∈ (A ∩ B) or x ∩ (A\B)
- ... in both cases
- ... then x ∈ A and x ∈ B, so x ∈ A ∩ B
- x !∈ B
- x ∈ A and x !∈ B, so x ∈ A\B
- x ∈ A and x !∈ B, so x ∈ A\B
- x ∈ B
- Either...
- Q.E.D.
- Show (A ∩ B) ∪ (A\B) ⊆ A
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