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With proper justification, state the negation of the following. (p ↔ q) v (~ q → ~ r) - Mathematics and Statistics

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Question

With proper justification, state the negation of the following.

(p ↔ q) v (~ q → ~ r)

Sum
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Solution

~[(p ↔ q) v (~ q → ~ r)]

≡ ~(p ↔ q) ˄ (~ q → ~ r)            ...[Negation of disjunction]

≡ [(p ˄ ~ q) v (q ∧ ~ p)] ∧ ~(~ q → ~ r)       ...[Negation of double implication]

≡ [(p ˄ ~ q) v (q ˄ ~ p)] ˄ [~ q ˄ ~(~ r)]      ...[Negation of implication]

≡ [(p ˄ ~ q) v (q ˄ ~ p)] ˄ (~ q ˄ r)           ...[Negation of negation]

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Chapter 1: Mathematical Logic - Exercise 1.8 [Page 21]
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