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Example Proof in Set Theory

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Senior Freshman Mathematics Mind Map on Example Proof in Set Theory, created by Luke Byrne on 22/04/2018.
Luke Byrne
Mind Map by Luke Byrne, updated more than 1 year ago
Luke Byrne
Created by Luke Byrne about 7 years ago
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Example Proof in Set Theory
  1. Proposition: "∀A, B sets, (A ∩ B) ∪ (A\B) = A".
    1. Show (A ∩ B) ∪ (A\B) ⊆ A
      1. ∀x ∈ (A ∩ B) ∪ (A\B), x ∈ (A ∩ B) or x ∈ A\B
        1. If x ∈ (A∩B), then clearly x ∈ A as A∩B ⊆ by definition.
          1. If x ∈ A\B, then by definition, x ∈ A and x !∈ B, so definitely x ∈ A.
            1. In both cases, x ∈ A as needed.
      2. Show A ⊆ (A ∩ B) ∪ (A\B)
        1. Either...
          1. x ∈ B
            1. ... then x ∈ A and x ∈ B, so x ∈ A ∩ B
              1. ... in both cases
                1. x ∈ (A ∩ B) or x ∩ (A\B)
                  1. so x ∈ (A ∩ B) ∪ (A\B), as needed
            2. x !∈ B
              1. x ∈ A and x !∈ B, so x ∈ A\B
          2. Q.E.D.
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