# Definition of conditional second set of unnecessary

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Chapter 1 / Exercise 58
Essential Calculus
Stewart
Expert Verified
(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) ≡ (? ∧ ¬?) ∨ ¬? ∨ ? (de Morgan) ≡ ((? ∧ ¬?) ∨ ¬?) ∨ ? (technically unnecessary () introduced to show how the absorption law applies) ≡ ¬? ∨ ? (absorption law) ≡ ? → ? (definition of conditional
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Chapter 1 / Exercise 58
Essential Calculus
Stewart
Expert Verified
2. Suppose two integers x and y are given. Formulate the contrapositive of the following statement, which is given in both verbal and symbolic form: If x is positive and y is non-negative, then xy is non-negative. ? > 0 ∧ ? ≥ 0 → ?? ≥ 0 . You must also give your answer in both forms. The contrapositive, in verbal form: If xy is negative, then x is non-positive, or y is negative. The contrapositive, in symbolic form: ?? < 0 → ? ≤ 0 ∨ ? < 0 3. Rewrite in standard “if.. then” form: .
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