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cs2102: Discrete Mathematics
Class 4: Logical Formulas
David Evans
University of Virginia
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Plan
Well-Ordering Principle Review
Logical Operations
Logical Formulas
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Odd Summation. Prove that for all n > 0, the sum of the first n odd numbers is n2.
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Odd Summation. Prove that for all n > 0, the sum of the first n odd numbers is n2.
(from Class 3)
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Odd Summation. Prove that for all n > 0, the sum of the first n odd numbers is n2.
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Odd Summation. Prove that for all n > 0, the sum of the first n odd numbers is n2.
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Well-Ordering Principle Proofs
Counter-examples
to P
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Logical Operations
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One-Input Boolean Operations
P NOT(P)
T F
F T
Are there any others?
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One-Input Boolean Operations
P IDENT(P)
T T
F F
Are there any others?
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One-Input Boolean Operations
P NOT(P) IDENT(P) TAUT(P) FALSE(P)
T F T T F
F T F T F
How many do we need to produce all of them?
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P NOT(P) IDENT(P) TAUT(P) FALSE(P)
T F T T F
F T F T F
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P NOT(P) IDENT(P) TAUT(P) FALSE(P)
T F T T F
F T F T F
Can we compute anything interesting with one-input Boolean operators?
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Two-Input Operators
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Two-Input Operators
P Q OP(P, Q)
T T
T F
F T
F F
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Two-Input Operators
P Q AND(P, Q) OR(P, Q) P IMPLIES Q P IFF Q XOR(P, Q)
T T T T
T F F T
F T F T
F F F F
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Two-Input Operators
P Q AND(P, Q) OR(P, Q) P IMPLIES Q P IFF Q XOR(P, Q)
T T T T T T F
T F F T F F T
F T F T T F T
F F F F T T F
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Implies and Causation
What is the relationship between P IMPLIES Q and P “CAUSES” Q?
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Two-Input Operators
P Q AND(P, Q) OR(P, Q) P IMPLIES Q P IFF Q XOR(P, Q)
T T T T T T F
T F F T F F T
F T F T T F T
F F F F T T F
How many two-input Boolean operators are there?
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P Q
AND
IFF
XOR
OR
IMPLIES
T T T T F F F T T T F F F T T T F F
T F T F T F F T F F T T F T T F T F
F T T F F T F F T F T F T T F T T F
F F T F F F T F F T F T T F T T T F
How many of these do we actually need?
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DeMorgan’s Laws
Augustus De Morgan (1806-1871)
We can replace OR with AND,
and replace AND with OR!
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Proof of equivalence:
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Proof of equivalence:
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Slack “quiz”: who of these knew De Morgan’s laws?
George
Boole
Ada,
Countess of
Lovelace
Thomas
Jefferson
Grace
Hopper
William of
Ockham
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Slack “quiz”: who of these knew De Morgan’s laws?
[Formal Logic, 1847]
George
Boole
(1815-1864)
Ada
Lovelace
(1815-1852)
Thomas
Jefferson
(1743-1826)
Grace
Hopper
(1906-1992)
William of
Ockham
(1285-1346)
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An Investigation of the Laws of Thought
on Which are Founded the Mathematical
Theories of Logic and Probabilities (1854)
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No content
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Completeness of Two-Input Operators
Is there something special about AND and OR?
Can we find an equivalence to AND that only uses NOT and XOR?
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P Q
AND
XOR
T T T F
T F F T
F T F T
F F F F
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P Q
AND
“FFTF”
T T T F
T F F F
F T F T
F F F F
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P Q
AND
“FFTF”
IFF
XOR
OR
IMPLIES
T T T T F F F T T T F F F T T T F F
T F T F T F F T F F T T F T T F T F
F T T F F T F F T F T F T T F T T F
F F T F F F T F F T F T T F T T T F
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Definitions
A formula is valid if there is no way to make it false.
A formula is satisfiable if there is some way to make it true.
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A formula is valid if there is no way to make it false.
A formula is satisfiable if there is some way to make it true.
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A formula is valid if there is no way to make it false.
A formula is satisfiable if there is some way to make it true.
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A formula is valid if there is no way to make it false.
A formula is satisfiable if there is some way to make it true.
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Charge
• If you need to enroll in the class, bring me a
course action form to sign
• Due Friday (6:29pm): PS1
• Finish reading MCS Ch 3 next week