Week 1 Homework Solution.pdf - MAT 243 Online Written...

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© 2017 R. Boerner, ASU School of Mathematical and Statistical Sciences.This solution is licensed to ASU students registered for MAT 243 classes for free personal use. It is not to be uploaded to third-party websites, especially not ones that sell such unlicensed content. If you found this document on a third-party website such as Course Hero or Chegg, the document is being served to you in violation of copyright law. (de Morgan) ≡ (? ∨ (¬? ∨ ?)) ∧ (¬? ∨ (¬? ∨ ?))(distributive law, associative law) ≡ (? ∨ (¬? ∨ ?)) ∧ ((¬? ∨ ¬?) ∨ ?)(associative law) ≡ ((¬? ∨ ?) ∨ ?) ∧ ((¬? ∨ ¬?) ∨ ?)(commutative law) ≡ (¬? ∨ (? ∨ ?)) ∧ ((¬? ∨ ¬?) ∨ ?)(associative law) ≡ (¬? ∨ ?) ∧ (¬? ∨ ?)(idempotent law) ≡ (¬? ∨ ?)(idempotent law) ≡ ? → ?(definition of conditional) This solution is tedious and hard to read due to repeated application of the commutative and associative laws. If we accept it as obvious that any multiple conjunction, or multiple disjunction, can be arbitrarily rearranged (and therefore also don’t need parentheses to indicate an order of operation) then a shorter and much more readable solution is possible: (? → ?) → (? → ?)≡ ¬(¬? ∨ ?) ∨ (¬? ∨ ?)(definition of conditional, second set of () unnecessary but written for clarity) ≡ (? ∧ ¬?) ∨ (¬? ∨ ?)(de Morgan, second set of () unnecessary but written for clarity) ≡ (? ∨ ¬? ∨ ?) ∧ (¬? ∨ ¬? ∨ ?)(distributive law) ≡ (¬? ∨ ?) ∧ (¬? ∨ ?)(idempotent law) ≡ (¬? ∨ ?)(idempotent law) ≡ ? → ?(definition of conditional). An even shorter solution is produced by just using the absorption law after de Morgan: (? → ?) → (? → ?)≡ ¬(¬? ∨ ?) ∨ (¬? ∨ ?)(definition of conditional) MAT 243 Online Written Homework Assignments for Week 1/15 - Solution1.Show that(? → ?) → (? → ?)is logically equivalent to ? → ?using logical equivalence rules. Name each rule. You will get no credit for any other type of solution, such as a solution by truth table. (? → ?) → (? → ?)≡ ¬(¬? ∨ ?) ∨ (¬? ∨ ?)(definition of conditional) ≡ (? ∧ ¬?) ∨ (¬? ∨ ?)

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