Last Update: 29 October 2010
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|A ∪ B| ≠ |A| + |B|
"‾" is an operator on sets that corresponds to (or is defined in terms of) "¬", which is an operator on propositions
proof:
‾(A ∪ B) | = {x | x ∈ ‾(A ∪ B)} | : definition of set |
= {x | x ∉ (A ∪ B)} | : definition of ‾ | |
= {x | ¬[x ∈ (A ∪ B)]} | : definition of ∉ | |
= {x | ¬[(x ∈ A) ∨ (x ∈ B)]} | : definition of ∪ | |
= {x | (x ∉ A) ∧ (x ∉ B)} | : DeM for ∨ & def of ∉ | |
= {x | (x ∈ ‾A) ∧ (x ∈ ‾B)} | : definition of ‾ | |
= {x | x ∈ (‾A ∩ ‾B)} | : definition of ∩ | |
= (‾A) ∩ (‾B) | : definition of set |
proof:
(S – T) ∩ (T – S) | = (S ∩ ‾T) ∩ (T ∩ ‾S) | : Lemma, based on def of ‾(*) |
= (S ∩ ‾S) ∩ (T ∩ ‾T) | : commutativity of ∩ | |
= ∅ ∩ ∅ | : complement law (p. 124) | |
= ∅ | : def of ∅ & ∩ |
(S – T) ∩ (T – S) | = {x | x ∈ [(S – T) ∩ (T – S)]} | : def of set |
= {x | [x ∈ (S – T)] ∧ [x ∈ (T – S)]} | : def of ∩ | |
= {x | (x ∈ S ∧ x ∉ T) ∧ (x ∈ T ∧ x ∉ S)} | : def of – | |
= {x | x ∈ S ∧ x ∉ S} ∩ {x | x ∈ T ∧ x ∉ T} | : commutativity of ∧ & def of ∩ | |
= ∅ ∩ ∅ | : def of ∅ | |
= ∅ | : def of ∅ & ∩ |
‾[A ∪ (B ∩ C)] | = ‾A ∩ ‾(B ∩ C) | : DeM |
= ‾A ∩ (‾B ∪ ‾C) | : DeM |
‾[A ∪ (B ∩ C)] | = {x | x ∉ [A ∪ (B ∩ C)]} | : (justifications are left to you!) |
= {x | ¬[x ∈ [A ∪ (B ∩ C)]]} | ||
= {x | ¬[(x ∈ A) ∨ (x ∈ (B ∩ C))]} | ||
= {x | ¬(x ∈ A) ∧ ¬(x ∈ (B ∩ C))} | ||
= {x | (x ∈ ‾A) ∧ (x ∈ ‾(B ∩ C))} | ||
= {x | (x ∈ ‾A) ∧ (x ∈ (‾B ∪ ‾C))} | ||
= {x | x ∈ [‾A ∩ (‾B ∪ ‾C)]} | ||
= ‾A ∩ (‾B ∪ ‾C) |
Notation: