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13397593
Example Proof in Set Theory
Description
Senior Freshman Mathematics Mind Map on Example Proof in Set Theory, created by Luke Byrne on 22/04/2018.
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proof
set theory
set theory proof
quantifiers
logic
mathematics
senior freshman
Mind Map by
Luke Byrne
, updated more than 1 year ago
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Created by
Luke Byrne
over 6 years ago
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Resource summary
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.
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
x !∈ B
x ∈ A and x !∈ B, so x ∈ A\B
Q.E.D.
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