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# Note for Discrete Mathematics - DMS By Kgs Gokul

• Discrete Mathematics - DMS
• Note
• Anna university - ACEW
• Computer Science Engineering
• 5 Topics
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UNIT - I PART-A Propositional Logic 1. Using the statements R: John is rich, H: John is happy, write βJohn is poor or he is both rich and unhappyβ in symbolic form. Solution: (M/J 2009) R: John is rich βΈπ: John is poor H: John is happy βΈπ»: John is unhappy The symbolic form is βΈπ β(πβ βΈπ») 2. Write the statement. βThe sun is bright and the humidity is not highβ in symbolic form. Solution: (M/J 2009) P: The sun is bright Q: The humidity is high βΈπ: The humidity is not high The symbolic form is π ββΈπ 3. What are the contrapositive, the converse, and the inverse of the conditional statement. βif you work hard then you will be rewardedβ Solution: (AU M/J 2013) P: You work hard βΈπ: You will not work hard Q: You will be rewarded βΈπ: You will not be rewarded CONVERSE: π β π .if you will be rewarded then you work hard CONTRAPOSITIVE: βΈπ β βΈπ .If you will not rewarded then you will not work hard INVERSE: βΈπ β βΈπ .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 βIf there is rain, then I buy an umbrellaβ. (AU N/D 2016) Solution: P: There is rain βΈπ: There is no rain Q: I buy an umbrella βΈπ: I do not buy an umbrella CONVERSE: π β π .If I buy umbrella then there is rain CONTRAPOSITIVE: βΈπ β βΈπ .If I do not buy a umbrella, then there is no rain. 5. Express the statement βGood food is not cheapβ in symbolic form. Solution: ( M/J 2008) Let P: food is good Q: food is cheap Good food is not cheap: π β βΈπ 6. Construct a truth table for the compound proposition (π β π) β (π β π). Solution: (N/D2013) πβπ πβπ (π β π) β (π β π) 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 β¨ βΈπ is a tautology. 2

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8. Is (βΈπ β§ (π β¨ π)) β π a tautology. (M/J 2014) Solution: 9. p q πβ¨π T T T T F F F βΈπ βΈp β§ (π β§ π) βΈp β§ (π β§ π) β π 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¨ βΈ(π· β§ πΈ) is a tautology. Solution: (AU M/J 2012) πβ§π βΈ(π β§ π) P β¨ βΈ(π β§ π) 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 (πΈ β πΉ)) β ((π· β πΈ) β (π· β πΉ)) is a tautology. Solution: (AU A/M2011) P β (π β π)) β ((π β π) β (π β π)) β ((π β π) β (π β π)) β ((π β π) β (π β π)) β βΈ((π β π) β (π β π)) β¨ ((π β π) β (π β π)) βπ 3

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11.Construct the truth table for the compound proposition (π β π) β (βΈπ β βΈπ) Solution: p [N/D 2014] βΈπ q βΈπ (π β π) β (βΈπ β βΈπ) πβπ βΈπ β βΈπ (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 π· β πΈ. Solution: [AU A/M 2017] π π πβπ T T T T F F F T T F F T 13.Give the truth table of π» β π»βπ­ [N/D 2015] Solution: π β πβπΉ βπβπΉβπΉ 14.Construct the truth table for π· β βΈπΈ. Solution: βΈπ [AU N/D 2016] π π π β βΈπ T T F F T F T T F T F T F F T T 4