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The Barber’s Puzzle
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The Barber’s Puzzle
In Puzzletown John has opened a new barber shop .
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The Barber’s Puzzle
In Puzzletown John has opened a new barber shop .
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The Barber’s Puzzle
In Puzzletown John has opened a new barber shop .
John shaves all men, and only those men, who do not shave
themselves.
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The Barber’s Puzzle
In Puzzletown John has opened a new barber shop .
John shaves all men, and only those men, who do not shave
themselves.
Who is shaving John the barber?
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Applied Logic:
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Applied Logic:
You come across 3 people, A, B and C.
one person is honest and always tells the truth
one is a notorious liar
one is a pokerface, sometimes liar sometimes honest
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Applied Logic:
You come across 3 people, A, B and C.
one person is honest and always tells the truth
one is a notorious liar
one is a pokerface, sometimes liar sometimes honest
They make the following statements:
A says: "I love mathematics."
B says: "C always tells the truth."
C says: "A hates math."
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Applied Logic:
You come across 3 people, A, B and C.
one person is honest and always tells the truth
one is a notorious liar
one is a pokerface, sometimes liar sometimes honest
They make the following statements:
A says: "I love mathematics."
B says: "C always tells the truth."
C says: "A hates math."
Who is most likely the honest one?
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Recall from last time:
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Recall from last time:
A statement is called a Proposition if it is either
true or false.
true ↔ truth value 1.
false ↔ truth value 0.
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Logical Operators and Truth Tables
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Logical Operators and Truth Tables
The operator "neg":
¬ or "
not"
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Logical Operators and Truth Tables
The operator "neg":
¬ or "
not"
"¬" changes proposition p to its opposite ¬p.
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Logical Operators and Truth Tables
The operator "neg":
¬ or "
not"
"¬" changes proposition p to its opposite ¬p.
"¬" reverses truth values.
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Logical Operators and Truth Tables
The operator "neg":
¬ or "
not"
"¬" changes proposition p to its opposite ¬p.
"¬" reverses truth values.
the proposition "3+2=6" if false, but the proposition
"¬(3 + 2 = 6)" is true.
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Logical Operators and Truth Tables
The operator "neg":
¬ or "
not"
"¬" changes proposition p to its opposite ¬p.
"¬" reverses truth values.
the proposition "3+2=6" if false, but the proposition
"¬(3 + 2 = 6)" is true.
"¬(3 + 2 = 6)" is not the same as "3 + 2 = 5".
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Logical Operators and Truth Tables
The operator "neg":
¬ or "
not"
"¬" changes proposition p to its opposite ¬p.
"¬" reverses truth values.
the proposition "3+2=6" if false, but the proposition
"¬(3 + 2 = 6)" is true.
"¬(3 + 2 = 6)" is not the same as "3 + 2 = 5".
the effect of "¬" is shown in the Truth Table:
p
¬p
1
0
0
1
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The Operators ∧ and ∨
∧ := ‘and’, ∨ :=‘or’
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The Operators ∧ and ∨
∧ := ‘and’, ∨ :=‘or’
used to combine elementary propositions
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The Operators ∧ and ∨
∧ := ‘and’, ∨ :=‘or’
used to combine elementary propositions
rules for truth values follow ‘sound common sense’:
p, q
p∧q
p∨q
1,1
1
1
1,0
0
1
0,1
0
1
0,0
0
0
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Examples
‘(5 ≥ 3) ∧ (5 ≥ 7)’ is false
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Examples
‘(5 ≥ 3) ∧ (5 ≥ 7)’ is false
‘(5 ≥ 3) ∨ (5 ≥ 7)’ is true.
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Examples
‘(5 ≥ 3) ∧ (5 ≥ 7)’ is false
‘(5 ≥ 3) ∨ (5 ≥ 7)’ is true.
‘¬p ∧ p’ is always false: a contradiction .
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Examples
‘(5 ≥ 3) ∧ (5 ≥ 7)’ is false
‘(5 ≥ 3) ∨ (5 ≥ 7)’ is true.
‘¬p ∧ p’ is always false: a contradiction .
‘¬p ∨ p’ is always true: a tautology.
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Theorem 1
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Theorem 1
‘¬(p ∧ q)’ is equivalent to ‘¬p ∨ ¬q’.
‘¬(p ∨ q)’ is equivalent to ‘¬p ∧ ¬q’.
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Theorem 1
‘¬(p ∧ q)’ is equivalent to ‘¬p ∨ ¬q’.
‘¬(p ∨ q)’ is equivalent to ‘¬p ∧ ¬q’.
Proof:
Check all possibilities in the following truth tables:
p, q
p∧q
¬(p ∧ q) ¬p ¬q
¬p ∨ ¬q
1,1
1
0
0
0
0
1,0
0
1
0
1
1
0,1
0
1
1
0
1
0,0
0
1
1
1
1
⋄
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Where are the errors in this Table?
p, q, r
(p ∧ q)
¬(p ∧ q)
¬(p ∧ q) ∨ r
1,1,1
1
0
1
1,1,0
1
0
1
1,0,1
0
1
1
0,1,1
0
1
1
0,1,0
0
1
1
0,0,1
0
1
1
0,0,0
0
1
1
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