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Transcript
Section 1.3 Truthtellers, liars and propositional logic We call a sentence a proposition p: q: r: A formal proposition 1. 2. Given propositions p and q, their conjunction, written p q is 3. Given propositions p and q, their disjunction, written p V q 4. Given a proposition p, the compound statement p Truth Tables bit operations corresponding to logical connectives 1010 1100 0011 1011 AND OR XOR Combining operations Two statements are logically equivalent Proposition 1 (p. 32) (DeMorgan’s Laws) Let p and q be any propositions. Then 1. (p V q) is logically equivalent to p q 2. (p q) is logically equivalent to p V q Example A tautology is A contradiction is Table of Equivalences Commutative Associative Distributive Identity Negation Double negative Idempotent DeMorgan's Laws Universal Absorption pqqp (p q) r = p (q r) p (q V r) (p q) V (p r) pTp p V p T ( p) p ppp (p q) p V q pVTT p (p V q) p pVqqVp (p V q) V r p V (q V r) p V (q r) (p V q) (p V r) pVFp p p F pVpp (p V q) p q pFF p V (p q) p Proof Example Show (p q) V q p V q Section 1.5 Implications An implication is Truth table for the implication p q T T T F F T F F pq The biconditional statement: Notation: p q. p q T T T F F T F F pq Terminology and Precedence Alternate Ways of Expressing Implications English Word Logical Connective Logical Expression and; but; also; moreover conjunction pq or disjunction pVq implication pq if p, then q p implies q p, therefore q p only if q q follows from p p is a sufficient condition for q q is a necessary condition for p p if and only if q equivalence pq p is necessary and sufficient for (biconditional) q not p it is false that p … it is not true that p … negation p Examples: The flowers bloom only if you water them. If it rains hard, it’s necessary to carry and umbrella. Having one even factor is sufficient to have an even product. Contrapositives, converses, and inverses Definition Consider the implication p q 1. The converse of the implication is 2. The inverse of the implication is 3. The contrapositive of the implication is Proposition 3 1. An implication and its contrapositive are logically equivalent 2. The converse and inverse of an implication are logically equivalent 3. The implication is not logically equivalent to its converse Showing p q and q p are logically equivalent and that p q and q p are not p q q p pq q p qp Using Equivalence in Proofs Statement: If n2 is odd then n is odd Contrapositive: Indirect (contrapositive) proof: Direct proof: Statement: If a and b are odd numbers, then a•b is odd. Inverse: Translating English sentences If the file is not damaged and the processor is fast, then the printer is slow. Translate: The program is efficient only if it executes fast and doesn’t have a bug. p: q: r: System Specifications Consistency Consistency Example 1. The system is in multiuser state iff it is operating normally. 2. If the system is operating normally, the kernel is functioning. 3. The kernel is not functioning or the system is in interrupt mode. 4. If the system is not in multiuser state then it is in interrupt mode. 5. The system is not in interrupt mode. p: q: r: s: