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Inference Rules and Proofs

Z: Inference Rules and Proofs

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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 systemtautologies--propositions which are always true (A = A)contradictions--propositions which are never true (A = not A)

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Logical Operators

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Inference Rule--Z Notation

Abbreviations:“intro” = introduction

“elim” = elimination

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AND Rules

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OR Rules

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IMPLICATION rules

(implication, equivalence)

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NEGATION Rules

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Truth Table Formulation

In terms of sets:

P

P

“universe”

P

Q

P Q

P Q

Q

P Q

P

QP

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Proof example: AND is commutative

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Proof example: OR is commutative

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Exercise: associativity

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Proof example: implication (1)

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Proof example: implication (2)

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Proof example: deMorgan’s Law

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Proof example: Law of the excluded middle