Logic Set Theory Quantification Unified

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

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Unformatted text preview: 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 )≡ N OT (∀ x ∈∅ : N OT P ( x )) ∀ x ∈∅ : N OT P ( x )≡ N OT (∃ x ∈∅ : P ( x )) ∀ x ∈∅ : N OT P ( x )≡ N OT ( FALSE ) N OT ( FALSE )≡ TRU E ∀ x ∈∅ : N OT P ( x )≡ TRU E 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 )≡ TRU E Implication (Logic) Implication is generally used when you must Quantify over a set (tying Logic and Set Theory together).Implication is generally used when you must Quantify over a set (tying Logic and Set Theory together)....
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This note was uploaded on 06/20/2011 for the course CSC CSC5921 taught by Professor Albert during the Spring '11 term at University of Colorado Denver.

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

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