Inference Rules and Proofs

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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
system tautologies--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
For n input variables, truth table would have 2n
rows using truth tables for expressions and
proofs is therefore not a practical or efficient
method of computation
In terms of sets
universe
P ? Q
P ? Q
P
Q
P
Q
P
P
? P
P ? Q
Q
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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 deMorgans Law
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Proof example Law of the excluded middle