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 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 Discrete Mathematics – Logic Equivalence 9-4 Logical Equivalence of Quantified Statements Two quantified statements are said to be logically equivalent if they are equivalent for any given universe. Consider statements 5 x (P(x) Q(x)) and ( 5 x P(x)) ( 5 x Q(x)) We prove that they are NOT logically equivalent. Discrete Mathematics – Logic Equivalence 9-5 Logical Equivalence of Quantified Statements (cntd) We have to find a universe, in which they are not equivalent. Let the universe consist of integers, P(x) means x > 5, and Q(x) means x < 3. Then ( 5 x P(x)) ( 5 x Q(x)) claims that ``there is a number greater than 5, and there is a number less than 3’’. This is true, as 6 witnesses the first claim, and 2 - the second claim. 5
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09h - Discrete Mathematics Logic Equivalence 9-2 Previous...

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