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Logic Set Theory Quantification Unified

Logic Set Theory Quantification Unified - Logic Set Theory...

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Logic Set Theory Quantification Unified Logic is Logic: Boolean Truth Values Set Theory is Set Theory: Manipulation of Sets Quantification is a way to tie Logic and Set Theory together. Special Case: The Empty Set The Empty Set contains no elements. Because it is a Set, you can Quantify over it. x ∈∅ : P ( x )≡ FALSE P(x) is arbitrary (it doesn't matter what the predicate is). There are no elements in so no x ∈∅ can exist. x ∈∅ : P ( x )≡ FALSE x ∈∅ : P ( x )≡ NOT (∀ x ∈∅ : NOT P ( x )) x ∈∅ : NOT P ( x )≡ NOT (∃ x ∈∅ : P ( x )) x ∈∅ : NOT P ( x )≡ NOT ( FALSE ) NOT ( FALSE )≡ TRUE x ∈∅ : NOT P ( x )≡ TRUE P(x) is arbitrary (it doesn't matter what the predicate is). If P(x) is arbitrary so is NOT P(x). Two rules: x ∈∅ : P ( x )≡ FALSE x ∈∅ : P ( x )≡ TRUE
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Implication (Logic) Implication is generally used when you must Quantify over a set (tying Logic and Set Theory together).
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