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Note for Discrete Mathematics - DMS By Kgs Gokul

  • Discrete Mathematics - DMS
  • Note
  • Anna university - ACEW
  • Computer Science Engineering
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UNIT - I PART-A Propositional Logic 1. Using the statements R: John is rich, H: John is happy, write “John is poor or he is both rich and unhappy” in symbolic form. Solution: (M/J 2009) R: John is rich ℸ𝑅: John is poor H: John is happy ℸ𝐻: John is unhappy The symbolic form is ℸ𝑅 ⋁(𝑅⋀ ℸ𝐻) 2. Write the statement. “The sun is bright and the humidity is not high” in symbolic form. Solution: (M/J 2009) P: The sun is bright Q: The humidity is high ℸ𝑄: The humidity is not high The symbolic form is 𝑃 ⋀ℸ𝑄 3. What are the contrapositive, the converse, and the inverse of the conditional statement. “if you work hard then you will be rewarded” Solution: (AU M/J 2013) P: You work hard ℸ𝑃: You will not work hard Q: You will be rewarded ℸ𝑄: You will not be rewarded CONVERSE: 𝑄 → 𝑃 .if you will be rewarded then you work hard CONTRAPOSITIVE: ℸ𝑄 → ℸ𝑃 .If you will not rewarded then you will not work hard INVERSE: ℸ𝑃 → ℸ𝑄 .If you will not work hard then you will not be rewarded. 1

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4. Give the contra positive statement of the statement “If there is rain, then I buy an umbrella”. (AU N/D 2016) Solution: P: There is rain ℸ𝑃: There is no rain Q: I buy an umbrella ℸ𝑄: I do not buy an umbrella CONVERSE: 𝑄 → 𝑃 .If I buy umbrella then there is rain CONTRAPOSITIVE: ℸ𝑄 → ℸ𝑃 .If I do not buy a umbrella, then there is no rain. 5. Express the statement “Good food is not cheap” in symbolic form. Solution: ( M/J 2008) Let P: food is good Q: food is cheap Good food is not cheap: 𝑃 → ℸ𝑄 6. Construct a truth table for the compound proposition (𝒑 → 𝒒) → (𝒒 → 𝒑). Solution: (N/D2013) 𝑝→𝑞 𝑞→𝑝 (𝑝 → 𝑞) → (𝑞 → 𝑝) p q T T T T T T F F T T F T T F F F F T T T 7. Define Tautology with an example. Solution: (AUN/D 2012) A statement formula which is true always irrespective of the truth values of the individual variables is called a tautology. Eg: P ∨ ℸ𝑃 is a tautology. 2

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8. Is (ℸ𝒑 ∧ (𝒑 ∨ 𝒒)) → 𝒒 a tautology. (M/J 2014) Solution: 9. p q 𝑝∨𝑞 T T T T F F F ℸ𝑃 ℸp ∧ (𝑝 ∧ 𝑞) ℸp ∧ (𝑝 ∧ 𝑞) → 𝑞 F F T T F F T T T T T T F F T F T Using truth table show that the proposition P ∨ ℸ(𝑷 ∧ 𝑸) is a tautology. Solution: (AU M/J 2012) 𝑃∧𝑄 ℸ(𝑃 ∧ 𝑄) P ∨ ℸ(𝑃 ∧ 𝑄) P Q T T T F T T F F T T F T F T T F F F T T 10.Show that (P→ (𝑸 → 𝑹)) → ((𝑷 → 𝑸) → (𝑷 → 𝑹)) is a tautology. Solution: (AU A/M2011) P → (𝑄 → 𝑅)) → ((𝑃 → 𝑄) → (𝑃 → 𝑅)) ⇒ ((𝑃 → 𝑄) → (𝑃 → 𝑅)) → ((𝑃 → 𝑄) → (𝑃 → 𝑅)) ⇒ ℸ((𝑃 → 𝑄) → (𝑃 → 𝑅)) ∨ ((𝑃 → 𝑄) → (𝑃 → 𝑅)) ⇒𝑇 3

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11.Construct the truth table for the compound proposition (𝒑 → 𝒒) ↔ (ℸ𝒑 → ℸ𝒒) Solution: p [N/D 2014] ℸ𝑝 q ℸ𝑞 (𝑝 → 𝑞) ↔ (ℸ𝑝 → ℸ𝑞) 𝑝→𝑞 ℸ𝑝 → ℸ𝑞 (1) (2) (3) T T F F T T T T F F T F T F F T T F T F F F F T T T T T 12.Find the truth table for 𝑷 → 𝑸. Solution: [AU A/M 2017] 𝑝 𝑞 𝑝→𝒒 T T T T F F F T T F F T 13.Give the truth table of 𝑻 ↔ 𝑻⋀𝑭 [N/D 2015] Solution: 𝑇 ↔ 𝑇⋀𝐹 ⇔𝑇↔𝐹⇔𝐹 14.Construct the truth table for 𝑷 → ℸ𝑸. Solution: ℸ𝑞 [AU N/D 2016] 𝑝 𝑞 𝑝 → ℸ𝒒 T T F F T F T T F T F T F F T T 4

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