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Tutorial 1: Logic

Tutorial 1: Logic. Peter Poon. Self Introduction. You can call me Peter Email: cypoon@cse.cuhk.edu.hk Office: SHB117 Office hour: Friday 10:00am – 12:00 noon Topics responsible: Logic and proofs. Agenda. Proof Distributive Law Construct and simplify Contrapositive Story. Proof.

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Tutorial 1: Logic

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  1. Tutorial 1: Logic Peter Poon

  2. Self Introduction • You can call me Peter • Email: cypoon@cse.cuhk.edu.hk • Office: SHB117 • Office hour: Friday 10:00am – 12:00 noon • Topics responsible: Logic and proofs

  3. Agenda • Proof • Distributive Law • Construct and simplify • Contrapositive • Story

  4. Proof • How to prove two statement are logically equivalent / not equivalent? • Prove or disprove

  5. Proof • Use truth table or equivalence laws to prove

  6. Proof

  7. Distributive Law Like extracting common factor 2 * (3 + 5) = (2 * 3) + (2 * 5) Consider If p is true, If p is false, both L.H.S and R.H.S are false

  8. Construct and simplify • Construct and simplify the formulas of f(x, y, z)

  9. Construct and simplify • Construct and simplify the formulas of f(x, y, z) Very long!!!

  10. Construct and simplify • Construct and simplify the formulas of f(x, y, z) • We can find the opposite

  11. Construct and simplify • Simplify the formulas of f(x, y, z) De Morgan’s law Distribution Law Distribution Law Distribution Law Negation Law Distribution Law Negation Law

  12. Contrapositive • Sometime you may want the contrapositive form • Find out the contrapositive form of

  13. Contrapositive • Find out the contrapositive form of • Use De Morgan’s law to help • Ans:

  14. Story • A detective has interviewed four witnesses to a crime. From their stories, the detective has concluded that • (a) If the butler is telling the truth, then so is the cook. • (b) The cook and the gardener cannot both be telling the truth. • (c) The gardener and the handyman are not both lying. • (d) If the handyman is telling the truth then the cook is lying. • Deduce who MUST be lying? (There may be more than one liar.)

  15. Story • First, define the variable • There are four people • Butler, Cook, Gardener, Handyman • Let B be “Butler is telling the truth” C be “Cook is telling the truth” G be “Gardener is telling the truth” H be “Handyman is telling the truth”

  16. Story • Then, write down the expression • (a) If the butler is telling the truth, then so is the cook. • (b) The cook and the gardener cannot both be telling the truth. • (c) The gardener and the handyman are not both lying. • (d) If the handyman is telling the truth then the cook is lying.

  17. Story • Make some assumption • Eg If B is true • Since , C is true • Since , G is false • Since , H is true • Since , C is false (contradiction!!!) • So, • B must be false • and C must be false

  18. Story • How about G and H? • We can’t determine them • Eg G = True, H = False and G = false, H = True are both valid solution.

  19. You are visiting a town. • The people in the town either always tell the truth or always lie. • One day you ask help from one townsman. • He said: "Don't worry, I will help you if and only if I tell the truth." Should you feel happy?

  20. Defining variable and write down expression • Let P be “the townsman always tell the truth” Q be “the townsman will help you” He said: “I will help you if and only if I tell the truth."

  21. Case 1: P is true Since , so he will help you Case 2: P is false Since , so So he will not help you? NO!!!

  22. Case 2: P is false • Since he is lying, is false • Verify by truth table or negate • Since P is false, so Q is true • So he will help you. • Therefore, you should be happy.

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