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

• How to prove two statement are logically equivalent / not equivalent?

• Prove or disprove

Proof

• Use truth table or equivalence laws to prove

p q

T T T

T F T

F T F

F F F

Proof

p q rT T T T T

T T F T T

T F T T T

T F F F F

F T T F F

F T F F F

F F T F F

F F F F F

Distributive Law

Like extracting common factor2 * (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

Construct and simplify

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

T T F F

T F T T

T F F T

F T T T

F T F F

F F T T

F F F F

Construct and simplify

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

Very long!!!

x y z f(x, y, z)T T T T

T T F F

T F T T

T F F T

F T T T

F T F F

F F T T

F F F F

Construct and simplify

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

T T T T

T T F F

T F T T

T F F T

F T T T

F T F F

F F T T

F F F F

Construct and simplify

• Simplify the formulas of f(x, y, z)

De Morgan’s law

Distribution Law

Distribution Law

Distribution Law

Distribution Law

Negation Law

Negation Law

Contrapositive

• Sometime you may want the contrapositive form

• Find out the contrapositive form of

Contrapositive

• Find out the contrapositive form of

• Use De Morgan’s law to help• Ans:

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.)

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”

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.

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

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.

G H

T F T T T T

F T T T T T

• 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?

• 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."

Case 1: P is trueSince , so he will help you

Case 2: P is falseSince , so So he will not help you? NO!!!

• 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.