# Week 1 HW Solutions.pdf - MAT 243 Online Written Homework...

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© 2017 R. Boerner, ASU School of Mathematical and Statistical Sciences MAT 243 Online Written Homework Assignments for Week 1 (units 1-3) - Solutions 1. 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) ≡ (? ∧ ¬?) ∨ (¬? ∨ ?) (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: (? → ?) → (? → ?) ≡ ¬(¬? ∨ ?) ∨ (¬? ∨ ?)