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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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IMPLICATION rules
(implication, equivalence)
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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