tutorial 1: logic
DESCRIPTION
Tutorial 1: Logic. Peter Poon. Self Introduction. You can call me Peter Email: [email protected] 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. - PowerPoint PPT PresentationTRANSCRIPT
Tutorial 1: Logic
Peter Poon
Self Introduction
• You can call me Peter• Email: [email protected]• 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.