CS173
Discrete Mathematical Structures
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Disjunctive Syllogism - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics
During the study of discrete mathematics, I found this course very informative and applicable.The main points in these lecture slides are:Disjunctive Syllogism, Proofs Simplification, Inference Rule, Tautology, Hypothetical Syllogism, Modus Ponens, Contrapositive, Direct Proofs, Proof Techniques, Valid Arguments, Proofs Fallacies, Denying Hypothesis
Typology: Slides
2012/2013
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Discrete Mathematical Structures
Proofs - Simplification
I am not a great skater and you are sleepy.
∴ you are sleepy.
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p ∧ q
∴ p
Tautology:
(p ∧ q) → p
Inference Rule: Simplification
Proofs - Hypothetical Syllogism
If you are an athlete, you are always hungry. If you are always hungry, you have a snickers in your backpack. ∴ If you are an athlete, you have a snickers in your backpack.
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p → q q → r
∴ p → r
Tautology:
((p → q) ∧ (q → r)) → (p → r)
Inference Rule: Hypothetical Syllogism
Proofs - A little quiz…
Amy is a computer science major.
∴ Amy is a math major or a computer science major.
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Addition
If Ernie is a math major then Ernie is geeky. Ernie is not geeky!
∴ Ernie is not a math major. Modus Tollens
Proofs - A little proof…
- M ∨ C Given
- ¬D → ¬C Given
- D → S Given
- ¬M Given
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5. C DS (1,4)
6. D MT (2,5)
7. S MP (3,6)
Ellen is smart!
Proofs - A little proof…
- M ∨ C Given
- ¬D → ¬C Given
- D → S Given
- ¬M Given
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- C Disjunctive Syllogism (1,4)
- C → D Contrapositive of 2
- C → S Hypothetical Syllogism (6,3)
- S Modus Ponens (5,7)
Ellen is smart!
Proof Techniques - direct proofs
A totally different example: Prove that if n = 3 mod 4, then n 2 = 1 mod 4.
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If n = 3 mod 4, then n = 4k + 3 for some int k.
But then, (^) n 2 = (4k + 3)(4k + 3)
= 16k 2 + 24k + 9 = 16k 2 + 24k + 8 + 1 = 4(4k 2 + 6k + 2) + 1 = 4j + 1 for some int j = 1 mod 4.
Proofs - Fallacies
Rules of inference, appropriately applied give
valid arguments.
Mistakes in applying rules of inference are
called fallacies.
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Proofs - valid arg or fallacy?
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If it rains then it is cloudy. It does not rain.
∴ It is not cloudy February!
If it is a car, then it has 4 wheels.
It is not a car.
∴ It doesn’t have 4 wheels. ATV
Denying the hypothesis.
((p → q) ∧ ¬p) → ¬q Not a tautology.