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Discrete Mathematics

by Kgs GokulKgs Gokul
Type: NoteInstitute: Anna university Specialization: Computer Science EngineeringOffline Downloads: 34Views: 552Uploaded: 1 month ago

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Kgs Gokul
Kgs Gokul
UNIT - I PART-A Propositional Logic 1. Using the statements R: John is rich, H: John is happy, write β€œJohn 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. β€œThe 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. β€œif 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 β†’ 𝑃 .if you will be rewarded then you work hard CONTRAPOSITIVE: ℸ𝑄 β†’ ℸ𝑃 .If you will not rewarded then you will not work hard INVERSE: ℸ𝑃 β†’ ℸ𝑄 .If you will not work hard then you will not be rewarded. 1
4. Give the contra positive statement of the statement β€œIf 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 β†’ 𝑃 .If I buy umbrella then there is rain CONTRAPOSITIVE: ℸ𝑄 β†’ ℸ𝑃 .If I do not buy a umbrella, then there is no rain. 5. Express the statement β€œGood 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 β†’ ℸ𝑄 6. Construct a truth table for the compound proposition (𝒑 β†’ 𝒒) β†’ (𝒒 β†’ 𝒑). Solution: (N/D2013) π‘β†’π‘ž π‘žβ†’π‘ (𝑝 β†’ π‘ž) β†’ (π‘ž β†’ 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
8. Is (ℸ𝒑 ∧ (𝒑 ∨ 𝒒)) β†’ 𝒒 a tautology. (M/J 2014) Solution: 9. p q π‘βˆ¨π‘ž T T T T F F F ℸ𝑃 β„Έp ∧ (𝑝 ∧ π‘ž) β„Έp ∧ (𝑝 ∧ π‘ž) β†’ π‘ž 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 ∨ β„Έ(𝑷 ∧ 𝑸) is a tautology. Solution: (AU M/J 2012) π‘ƒβˆ§π‘„ β„Έ(𝑃 ∧ 𝑄) P ∨ β„Έ(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 (Pβ†’ (𝑸 β†’ 𝑹)) β†’ ((𝑷 β†’ 𝑸) β†’ (𝑷 β†’ 𝑹)) is a tautology. Solution: (AU A/M2011) P β†’ (𝑄 β†’ 𝑅)) β†’ ((𝑃 β†’ 𝑄) β†’ (𝑃 β†’ 𝑅)) β‡’ ((𝑃 β†’ 𝑄) β†’ (𝑃 β†’ 𝑅)) β†’ ((𝑃 β†’ 𝑄) β†’ (𝑃 β†’ 𝑅)) β‡’ β„Έ((𝑃 β†’ 𝑄) β†’ (𝑃 β†’ 𝑅)) ∨ ((𝑃 β†’ 𝑄) β†’ (𝑃 β†’ 𝑅)) ⇒𝑇 3
11.Construct the truth table for the compound proposition (𝒑 β†’ 𝒒) ↔ (ℸ𝒑 β†’ ℸ𝒒) Solution: p [N/D 2014] ℸ𝑝 q β„Έπ‘ž (𝑝 β†’ π‘ž) ↔ (ℸ𝑝 β†’ β„Έπ‘ž) π‘β†’π‘ž ℸ𝑝 β†’ β„Έπ‘ž (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 β†’ 𝑸. Solution: [AU A/M 2017] 𝑝 π‘ž 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 β†’ ℸ𝑸. Solution: β„Έπ‘ž [AU N/D 2016] 𝑝 π‘ž 𝑝 β†’ ℸ𝒒 T T F F T F T T F T F T F F T T 4

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