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The graph of function given above m p the truth table

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Unformatted text preview: nnective. The graph of ) function ) given above M↔ (P, the truth table of the negation connective. Similarly, we associate to a connective ff f f f t t X in the set {∧, ∨, ⊕, →, ↔} a binary function MX : B ×tB → B. The truth ft f t t f tables of these connectives are t follows: t as tf f f f t P t Q M∧tP, Q) M∨ (P, Q) M⊕ (P, Q) M→ (P, Q) M↔ (P, Q) t f t t ( ff f f f t t You should tvery carefully inspect this table! It is t critical that you memorize f f t t f and fully understand the meaning of each connective. tf f t t f f The t t semantics of the language Propf is given by assigning ttruth values to t t t each proposition in Prop. Clearly, an arbitrary assignment of truth values is You should very carefully inspect everything is critical that with the meaning not interesting, since we would likethis table! Itto be consistentyou memorize 4 Conditional Perhaps the most important logical connective is the conditional, also known as implication: p -> q The statement asserts that q holds on the condition that p holds. We call p the hypothesis or premise, and q the conclusion or consequence. Typical usage in proofs: “If p, then q”; “p implies q”; “q when p”; “q follows from p” “p is sufficient for q”; “a sufficient condition for q is p”; “a necessary condition for p is q”; “q is necessary for p” 5 Logical Equivalence Two statements involving quantifiers and predicates are logically equivalent if and only if they have the same truth values no matter which predicates are substituted into these statements and which domain is used. We write A ≡ B for logically equivalent A and B. You use logical equivalences to derive more convenient forms of statements. Example: De Morgan’s laws. 6 De Morgan’s Laws ¬∀xP (x) ≡ ∃x¬P (x) ¬∃xP (x) ≡ ∀x¬P (x) ¬ ( p ∧ q ) ≡ ¬p ∨ ¬ q ¬ ( p ∨ q ) ≡ ¬p ∧ ¬ q 7 Valid Arguments An argument in propositional logic is a sequence of propositions that end with a proposition called conclusion. The argument is called valid if the conclusion follows from...
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