Podcast
Questions and Answers
What is the abbreviation for the Commutative rule?
What is the abbreviation for the Commutative rule?
- assoc
- dn
- comm (correct)
- imp
(R V S) V Q is equivalent to R V (S V Q).
(R V S) V Q is equivalent to R V (S V Q).
True (A)
What does De Morgan's Law state for the expression (R V S)?
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.
R → S is equivalent to R' V S.
What rule allows you to derive S from R and R → S?
What rule allows you to derive S from R and R → S?
What is the abbreviation for Hypothetical Syllogism?
What is the abbreviation for Hypothetical Syllogism?
P → Q and Q' → P' are examples of Contraposition.
P → Q and Q' → P' are examples of Contraposition.
(∀x)P(x) can derive _____ with t as a variable.
(∀x)P(x) can derive _____ with t as a variable.
What must be true for a constant symbol 'a' in Existential Instantiation?
What must be true for a constant symbol 'a' in Existential Instantiation?
Which rule provides a way to go from P(x) to (∀x)P(x)?
Which rule provides a way to go from P(x) to (∀x)P(x)?
Flashcards are hidden until you start studying
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).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.