2010_lec9 - CS245 - Winter 2010, Lecture 9 Shai Ben-David...

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Unformatted text preview: CS245 - Winter 2010, Lecture 9 Shai Ben-David Syntax vs. Semantics Syntax (form) Semantics (meaning) Defining propositional formulas. ”Which sequences of characters are legal propositions?” Example: Does ( a ∧ b ) = ( b ∧ a )? (no) Truth tables Logical Equivalence Example: Does ( a ∧ b ) ≡ ( b ∧ a )? (yes) Tautologies Definition: The ”=” symbol represents syntactic equality , which is essentially a character-by-character comparison. Definition: The ” ≡ ” symbol represents semantic equivalence , which indicates that both sides of the equation have the same meaning. Formal Proofs in Propositional Calculus We need proofs to know if statements are true. In proposition calculus, it is not necessary to use proofs to answer questions like “is ( p → ( q → p )) a tautology,” since truth tables are sufficient. So... why bother? Just as playing Monopoly is practice for real life scenarios in a simpler, less risky environment, propositional calculus proofs are practice for those involving “real” math. They demonstrate important principlespractice for those involving “real” math....
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This note was uploaded on 02/11/2010 for the course ART AFM101 taught by Professor Mr.lushman during the Spring '10 term at University of Toronto- Toronto.

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2010_lec9 - CS245 - Winter 2010, Lecture 9 Shai Ben-David...

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