Questions and Answers
What is the abbreviation for the Commutative rule?
(R V S) V Q is equivalent to R V (S V Q).
True
What does De Morgan's Law state for the expression (R V S)?
(R V S)' = R' Λ S'
R → S is equivalent to R' V S.
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What rule allows you to derive S from R and R → S?
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What is the abbreviation for Hypothetical Syllogism?
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P → Q and Q' → P' are examples of Contraposition.
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(∀x)P(x) can derive _____ with t as a variable.
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What must be true for a constant symbol 'a' in Existential Instantiation?
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Which rule provides a way to go from P(x) to (∀x)P(x)?
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Study Notes
Equivalence Rules
- RVS is equivalent to SVR (Commutative - comm).
- RΛS is equivalent to SΛR (Commutative).
- (R V S) V Q is equivalent to R V (S V Q) (Associative - assoc).
- (R Λ S) Λ Q is equivalent to R Λ (S Λ Q) (Associative).
- De Morgan’s Laws state that (R V S)′ is equivalent to R′ Λ S′ and (R Λ S)′ is equivalent to R′ V S′.
- R→S translates to R′ V S (Implication - imp).
- The double negation states R is equivalent to (R′)′ (Double negation - dn).
- P↔Q is defined as (P → Q) Λ (Q → P) (Equivalence - equ).
Inference Rules
- From R and R → S, S can be derived (Modus ponens - mp).
- From R → S and S′, R′ can be derived (Modus tollens - mt).
- R and S lead to RΛS (Conjunction - con).
- RΛS allows simplification to R or S (Simplification - sim).
- From R, RVS can be derived (Addition - add).
Additional Inference Rules
- P → Q and Q → R allow derivation of P→R (Hypothetical syllogism - hs).
- From P V Q and P′, Q can be derived (Disjunctive syllogism - ds).
- Contraposition allows the transformation of P→Q to Q′ → P′ (Contraposition - cont).
- Self-reference: from P, one can derive PΛP (Self-reference - self).
- For (P Λ Q) → R, the equivalent form is P → (Q → R) (Exportation - exp).
- P and P′ lead to Q (Inconsistency - inc).
- Distributive laws: P Λ (Q V R) is equivalent to (P Λ Q) V (P Λ R) and P V (Q Λ R) equals (P V Q) Λ (P V R).
Inference Rules – Instantiation/Generalization for Quantifiers
- Universal Instantiation allows P(t) to be derived from (∀x)P(x), where t is a variable or constant symbol, ensuring t is not within a quantifier's scope.
- Existential Instantiation infers P(a) from (∃x)P(x) using a constant symbol not already referenced in the proof sequence.
- Universal Generalization states that if P(x) is true, then (∀x)P(x) holds, under specific conditions about free variables.
- Existential Generalization allows deriving (∃x)P(x) from P(x) or P(a), ensuring x does not appear in P(a).
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Description
This quiz covers Equivalence Rules essential for CSE 2315. Students will review various logical expressions and their equivalent forms as well as related abbreviations. Understanding these concepts is crucial for success in the course.
