Discrete Mathematics 4 questions (truth tables)

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Discrete Mathematics (MAT 214) Homework 2 Complete the following problems on a separate sheet of paper. Write as much as possible for full credit, including calculator work. This assignment is due on Wednesday, January 30th . 1) Write negations for each of the following statements. (Assume that all variables represent fixed quantities or entities, as appropriate.) a) If today is New Year’s Eve, then tomorrow is January. b) If the decimal expansion of r is terminating, then r is rational. c) If x is nonnegative, then x is positive or x is 0. d) If n is divisible by 6, then n is divisible by 2 and n is divisible by 3. 2) Determine whether the following statement forms and logically equivalent: a) p → (q → r) and (p → q) → r b) p ↔ q and (∼p ∨ q) ∧ (∼q ∨ p) 3) Use truth tables to determine whether the following argument forms are valid. Indicate which columns represent the premises and which represent the conclusions, and include a sentence explaining how the truth table supports your answer. Your explanation should show that you understand what it means for a form of argument to be valid or invalid. a) ∴ (p ∧ q) → ∼r p ∨ ∼q ∼q → p ∼r b) p→r q→p ∴ (p ∨ q) → r 4) The following arguments are given. Using the valid argument forms from class (Modus Ponens, Modus Tollens, Generalization, ...), which can also be found on page 61 in the book, deduce the given conclusion from the premises, giving a reason for each step. a) p∨q q→r (p ∧ s) → t ∼r ∼q → (u ∧ s) ∴ t b) p→q r∨s ∼s → ∼t ∼q ∨ s ∼s (∼p ∧ r) → u w∨t ∴ u∧w

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