* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download HW-04 due 02/10
Axiom of reducibility wikipedia , lookup
History of logic wikipedia , lookup
Mathematical logic wikipedia , lookup
Propositional calculus wikipedia , lookup
Mathematical proof wikipedia , lookup
Intuitionistic logic wikipedia , lookup
Interpretation (logic) wikipedia , lookup
New riddle of induction wikipedia , lookup
Natural deduction wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Analytic–synthetic distinction wikipedia , lookup
Curry–Howard correspondence wikipedia , lookup
Combinatory logic wikipedia , lookup
Truth-bearer wikipedia , lookup
Laws of Form wikipedia , lookup
CmSc180 Discrete Mathematics Homework 04 due 02/10 1. For the implication ~P → Q indicate which of the following expressions is its contrapositive, its converse and its inverse (underline the correct one): Contrapositive: P → Q, ~Q → P, P → ~Q, ~P → ~Q, ~Q → ~P, Q → ~P Inverse: P → Q, ~Q → P, P → ~Q, ~P → ~Q, ~Q → ~P, Q → ~P Converse: P → Q, ~Q → P, P → ~Q, ~P → ~Q, ~Q → ~P, Q → ~P 2. Let P , Q, and R be the propositions P: I am awake Q: I work hard R: I dream of home Represent each of the following sentences as logical expressions: a. I dream of home only if I am not working hard b. Working hard is sufficient for me not to dream of home c. Being awake is necessary for me to work hard 3. Give direct proof of the following statement: For all n, if n is even then (n-1)(n+1) is odd 4. Prove by mathematical induction 12 + 22 + 32 + 42 + 52 + …. + n 2 = n(n+1)(2n+1)/6 5. Using the predicates class(x), difficult (x), boring(x) and appropriate quantifiers (,), represent in predicate logic the following sentences, write the negation of the predicate expression and translate back to English a. Some classes are difficult and boring. b. Difficult classes are not boring. c. No classes are difficult and boring 1 6. Represent the following arguments in predicate logic and determine whether they are valid or invalid. If valid determine the type of argument. If invalid determine the type of error if the type is known (converse or inverse error). Every adult is eligible to vote. John is eligible to vote. John is an adult. Some students play football John is a student. John plays football 2