1 Inference Rules and Proofs Z: Inference Rules and Proofs

2 Propositional logic The Z methodology is based on propositional logic basic operators of propositional logic: conjunction (AND); disjunction (OR); implication (  ); equivalence (  ) ; negation (NOT, ~) propositions--statements about the system tautologies--propositions which are always true (A = A) contradictions--propositions which are never true (A = not A)

3 Logical Operators

4 Inference Rule--Z Notation Abbreviations:“intro” = introduction “elim” = elimination

5 AND Rules

6 OR Rules

7 IMPLICATION rules (implication, equivalence)

8 NEGATION Rules

9 Truth Table Formulation In terms of sets: P P “universe”  P Q P  Q P  Q Q P  Q P QP For n input variables, truth table would have 2 n rows; using truth tables for expressions and proofs is therefore not a practical or efficient method of computation

10 Proof example: AND is commutative

11 Proof example: OR is commutative

12 Exercise: associativity

13 Proof example: implication (1)

14 Proof example: implication (2)

15 Proof example: deMorgan’s Law

16 Proof example: Law of the excluded middle