csci2110 tutorial 9: propositional logic
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CSCI2110 Tutorial 9: Propositional Logic. Chow Chi Wang ( cwchow ‘at’ cse.cuhk.edu.hk). Propositional Logic. I am lying right now…. Is he telling the truth or lying?. Propositional Statement. A Statement is a sentence that is either True or False . - PowerPoint PPT PresentationTRANSCRIPT
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CSCI2110 Tutorial 9:Propositional Logic
Chow Chi Wang (cwchow ‘at’ cse.cuhk.edu.hk)
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Propositional Logic
I am lying right now…
Is he telling the truth or lying?
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Propositional Statement• A Statement is a sentence that is either True or False.
• Which of these are propositional statements?1. The sun is shining.2. 3 + 4 = 7.3. It rained this morning.4. .5. for .6. Is it raining?7. Come to tutorial!8. 2012 is a prime number.9. Hello world!10. Discrete Maths is very interesting.
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Basic Logic Operators
P PT F
F T
, ~ (NOT) – Truth Table
∧ (AND) – Truth TableP Q PQT T T
T F FF T FF F F
∨ (OR) – Truth TableP Q PQT T T
T F TF T TF F F
• We can construct more complicated statements from these basic operators. (e.g. )
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XOR (Exclusive-OR)• When OR is used in its exclusive sense, “p xor q” means
“p or q but not both”.
• How to express XOR in terms of AND, OR and NOT?
(XOR) – Truth TableP Q PQT T F
T F TF T TF F F
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Writing Logical Formula for a Truth Table
• Idea 1: Look at the true rows and take the OR.
• For each true row:• Construct a clause that is an AND of the true variables.
• Take the OR of all the clauses obtained from the for loop.
P Q fT T FT F TF T FF F T
𝑃∧ 𝑄
𝑃∧ 𝑄(𝑃∧ 𝑄 )∨( 𝑃∧ 𝑄)
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Writing Logical Formula for a Truth Table
• Idea 2: Look at the false rows and take the AND.
• For each false row:• Construct a clause that is an AND of the true variables.
• Take the AND of all the negated clauses obtained from the for loop.
P Q fT T FT F TF T FF F T
(𝑃∧𝑄)
( 𝑃∧𝑄)~
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Logical Equivalence• Two statements are logically equivalent if they have the
same truth table.
• e.g.
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Tautology, Contradiction• A tautology is a statement that is always true.
• ,
• A contradiction is a statement that is always false.• ,
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Summary of Basic Logical Rules
Commutative laws:
Associative laws:
Distributive laws:
Identity laws:
Negation laws:
Double negative law:
Idempotent laws:
Universal bound laws:
De Morgan’s laws:
Absorption laws:
= tautology, = contradiction
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Simplifying Statements
(Distribution law)(Idempotent law)
(Identity law)
~(De Morgan’s law)
(Distributive law)(Idempotent law)
(Identity law)
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Conditional StatementIf P then Q
p is called the hypothesis; q is called the conclusion
P implies Q
P QT T T
T F FF T TF F T
𝑃→𝑄
– Truth Table
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Conditional Statement• is equivalent to .
• The contrapositive of “” is “”.
• (Statement) Every CUHK student has CULINK.• (Contrapositive) If someone doesn’t have CULINK, then
(s)he is not a CUHK student.
Important factA conditional statement is logically equivalent to its contrapositive.
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Argument• An argument is a sequence of statements.
• All statements but the final one are called assumptions or hypothesis.
• The final statement is called the conclusion.
• An argument is valid if whenever all the assumptions are true, then
the conclusion is true.
If today is Sunday, then it is holiday. Today is Sunday. Today is holiday.
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Valid Argument?
T T T T T T
T T F T F F
T F T F T T
T F F T T F
F T T T F T
F T F T F T
F F T T T T
F F F T T T
Invalid argument!
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Valid Argument?
T T T T T
T T F T F
T F T F T
T F F F T
F T T T T
F T F T F
F F T T T
F F F T T
Valid argument!
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Knights and Knaves
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Knights and KnavesSuppose you are visiting an island containing two types of people – knights and knaves.
Two natives and address you as follow:
: Both of us are knights.: is a knave.
What are and ?
Knights always tell the truth.Knaves always lie.
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Knights and Knaves: Both of us are knights.: is a knave.
Knights always tell the truth.Knaves always lie.
If is a knaveÞ is a knightÞ Both of them are knightsÞ In particular, is a knight Contradiction!
Therefore must be a knightÞ is a knave
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Knights and KnavesAnother two natives and approach you but only speaks.
: Both of us are knaves.
What are and ?
Knights always tell the truth.Knaves always lie.
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Knights and Knaves
: Both of us are knaves.
Knights always tell the truth.Knaves always lie.
If is a knightÞ Both of them are knavesÞ In particular, is a knaveContradiction!
Therefore must be a knaveÞ One of them is a knightÞ is a knight
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Knights and KnavesYou then encounter natives and .
: is a knave.: is a knave.
How many knaves are there?
Knights always tell the truth.Knaves always lie.
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Knights and Knaves: is a knave.: is a knave.
Knights always tell the truth.Knaves always lie.
If is a knightÞ is a knaveÞ is a knightÞ There is a knight and a knave
Similarly, if is a knightÞ is a knaveÞ is a knightÞ There is a knight and a knave
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Knights and KnavesFinally, you meet a group of six natives, and , who speak to you as follow:
: None of us is a knight.: At least three of us are knights.: At most three of us are knights.: Exactly five of us are knights.: Exactly two of us are knights.: Exactly one of us is a knight.
Knights always tell the truth.Knaves always lie.
Which are knights and which are knaves?
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Knights and Knaves: None of us is a knight.: At least three of us are knights.: At most three of us are knights.: Exactly five of us are knights.: Exactly two of us are knights.: Exactly one of us is a knight.
Knights always tell the truth.Knaves always lie.
Suppose is a knightÞ None of them is a knightÞ is a knaveContradiction!
So can only be a knave.
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Knights and Knaves: None of us is a knight.: At least three of us are knights.: At most three of us are knights.: Exactly five of us are knights.: Exactly two of us are knights.: Exactly one of us is a knight.
Knights always tell the truth.Knaves always lie.
Suppose is a knightÞ Exactly five of them are knightsÞ are knights (because at this
stage we already know is a knave)
Þ In particular, is a knightÞ Exactly one of them is a knightContradiction!
So can only be a knave. What we know at this stage:• is a knave
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Knights and Knaves: None of us is a knight.: At least three of us are knights.: At most three of us are knights.: Exactly five of us are knights.: Exactly two of us are knights.: Exactly one of us is a knight.
Knights always tell the truth.Knaves always lie.
Suppose is a knightÞ At least three of them can be
knightsÞ At least three people among
are knightsÞ Either or must be a knight (by
pigeon-hole principle)Þ Exactly one(or two) of them is a
knightContradiction!
So can only be a knave.
What we know at this stage:• is a knave• is a knave
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Knights and Knaves: None of us is a knight.: At least three of us are knights.: At most three of us are knights.: Exactly five of us are knights.: Exactly two of us are knights.: Exactly one of us is a knight.
Knights always tell the truth.Knaves always lie.
At this stage we already identify three people as knavesÞ There can be at most three of
them are knightsÞ is a knight
What we know at this stage:• is a knave• is a knave• is a knave
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Knights and Knaves: None of us is a knight.: At least three of us are knights.: At most three of us are knights.: Exactly five of us are knights.: Exactly two of us are knights.: Exactly one of us is a knight.
Knights always tell the truth.Knaves always lie.
Suppose is a knightÞ Exactly one of them is a knightContradiction!(because both are knights in this case)
Therefore must be a knave.
What we know at this stage:• is a knave• is a knave• is a knight• is a knave
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Knights and Knaves: None of us is a knight.: At least three of us are knights.: At most three of us are knights.: Exactly five of us are knights.: Exactly two of us are knights.: Exactly one of us is a knight.
Knights always tell the truth.Knaves always lie.
Suppose is a knaveÞ is the only knightÞ There is exactly one knightÞ is a knightContradiction!
Therefore must be a knight.
What we know at this stage:• is a knave• is a knave• is a knight• is a knave• is a knave
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Knights and Knaves: None of us is a knight.: At least three of us are knights.: At most three of us are knights.: Exactly five of us are knights.: Exactly two of us are knights.: Exactly one of us is a knight.
Knights always tell the truth.Knaves always lie.
Therefore:
• is a knave• is a knave• is a knight• is a knave• is a knight• is a knave
What we know at this stage:• is a knave• is a knave• is a knight• is a knave• is a knave
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Summary• How to write logical formula from a truth table?
• How to check whether two logical formulas are equivalent?
• How to simplify logical formula using rules including (but not limited to) De Morgan’s law?
• What are conditional statements and contrapositive?
• How to check whether an argument is valid or not?
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Thank You!