09h - Discrete Mathematics Logic Equivalence 9-2 Previous...

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Introduction Logic Equivalence Discrete Mathematics Andrei Bulatov Discrete Mathematics – Logic Equivalence 9-2 Previous Lecture Free and bound variables Multiple quantifiers and logic connectives Definitions, rules, and theorems Discrete Mathematics – Logic Equivalence 9-3 Rules Predicates and quantifiers are (implicitly) present in rules and laws ``Everyone having income more that $20000 must file a tax report’’ P(x) - ``x has income more than $20000’’ Q(x) - ``x must file a tax report’’ 2200 x (P(x) Q(x)) Discrete Mathematics – Logic Equivalence 9-4 Theorems Every theorem involves predicates and quantifiers ``For every statement there is an equivalent CNF’’ C(x) - ``x is a CNF’’ 2200 x 5 y (C(y) (x y)) ``A parallelogram is a rectangle if all its angles are equal’’ R(x) - ``parallelogram x is a rectangle’’ A(x) - ``all angles of x are equal’’ 2200 x (A(x) R(x)) Discrete Mathematics – Logic Equivalence 9-5 Universe and Interpretation A logic statement is meaningless 2200 x P(x) It only makes sense if we specify a universe and a particular meaning of the predicate universe: animals P(x): x has horns universe: cars P(x): x is red universe: numbers P(x): x is even Interpretation Discrete Mathematics – Logic Equivalence 9-6 Logical Equivalence of Predicates Recall that two compound statements Φ and Ψ are logically equivalent ( Φ ⇔ Ψ ) if and only if Φ ↔ Ψ is a tautology. For predicates: Two predicates P(x) and Q(x) are logically equivalent in a given universe if and only if, for any value a from the universe statements P(a) and Q(a) are equivalent if and only if the statement 2200 x (P(x) Q(x)) is true in the given universe ``A parallelogram is a rectangle if and only if all its angles are equal’’ P(x) - ``x is a rectangle’’ Q(x) - ``all angles of x are equal’’ P(x) Q(x) in the universe of parallelograms
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Discrete Mathematics – Logic Equivalence 9-7 Logical Equivalence of Quantified Statements Two quantified statements are said to be logically equivalent if they are equivalent for any given universe. Consider statements
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09h - Discrete Mathematics Logic Equivalence 9-2 Previous...

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