Book of Proof - Amazon S3

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11. If A and B are sets in a universal set U, then A ∪ B = A ∩ B.
Proof. Just observe the following sequence of equalities.
A ∪ B = U − (A ∪ B) (def. of complement)
= {x : (x ∈ U) ∧ (x ∉ A ∪ B)} (def. of −)
= {x : (x ∈ U)∧ ∼ (x ∈ A ∪ B)}
= {x : (x ∈ U)∧ ∼ ((x ∈ A) ∨ (x ∈ B))} (def. of ∪)
= {x : (x ∈ U) ∧ (∼ (x ∈ A)∧ ∼ (x ∈ B))} (DeMorgan)
= {x : (x ∈ U) ∧ (x ∉ A) ∧ (x ∉ B)}
= {x : (x ∈ U) ∧ (x ∈ U) ∧ (x ∉ A) ∧ (x ∉ B)} (x ∈ U) = (x ∈ U) ∧ (x ∈ U)
= {x : ((x ∈ U) ∧ (x ∉ A)) ∧ ((x ∈ U) ∧ (x ∉ B))} (regroup)
= {x : (x ∈ U) ∧ (x ∉ A)} ∩ {x : (x ∈ U) ∧ (x ∉ B)} (def. of ∩)
= (U − A) ∩ (U − B) (def. of −)
= A ∩ B (def. of complement)
The proof is complete.
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13. If A,B and C are sets, then A − (B ∪ C) = (A − B) ∩ (A − C).
Proof. Just observe the following sequence of equalities.
A − (B ∪ C) = {x : (x ∈ A) ∧ (x ∉ B ∪ C)} (def. of −)
= {x : (x ∈ A)∧ ∼ (x ∈ B ∪ C)}
= {x : (x ∈ A)∧ ∼ ((x ∈ B) ∨ (x ∈ C))} (def. of ∪)
= {x : (x ∈ A) ∧ (∼ (x ∈ B)∧ ∼ (x ∈ C))} (DeMorgan)
= {x : (x ∈ A) ∧ (x ∉ B) ∧ (x ∉ C)}
= {x : (x ∈ A) ∧ (x ∈ A) ∧ (x ∉ B) ∧ (x ∉ C)} (x ∈ A) = (x ∈ A) ∧ (x ∈ A)
= {x : ((x ∈ A) ∧ (x ∉ B)) ∧ ((x ∈ A) ∧ (x ∉ C))} (regroup)
= {x : (x ∈ A) ∧ (x ∉ B)} ∩ {x : (x ∈ A) ∧ (x ∉ C)} (def. of ∩)
= (A − B) ∩ (A − C) (def. of −)
The proof is complete.
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15. If A,B and C are sets, then (A ∩ B) − C = (A − C) ∩ (B − C).
Proof. Just observe the following sequence of equalities.
(A ∩ B) − C = {x : (x ∈ A ∩ B) ∧ (x ∉ C)} (def. of −)
= {x : (x ∈ A) ∧ (x ∈ B) ∧ (x ∉ C)} (def. of ∩)
= {x : (x ∈ A) ∧ (x ∉ C) ∧ (x ∈ B) ∧ (x ∉ C)} (regroup)
= {x : ((x ∈ A) ∧ (x ∉ C)) ∧ ((x ∈ B) ∧ (x ∉ C))} (regroup)
= {x : (x ∈ A) ∧ (x ∉ C)} ∩ {x : (x ∈ B) ∧ (x ∉ C)} (def. of ∩)
= (A − C) ∩ (B − C) (def. of ∩)
The proof is complete.
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17. If A,B and C are sets, then A × (B ∩ C) = (A × B) ∩ (A × C).
Proof. See Example 8.12.
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