1. Lecture 1.4: Rules of Inference
CS 250, Discrete Structures, Fall 2015
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren

2. Lecture 1.4 - Rules of Inference
Course Admin
Slides from previous lectures all posted
Expect HW1 to be coming in around coming Monday
Questions?
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Lecture Rules of Inference

3. Lecture 1.4 - Rules of Inference
Outline
Rules of Inference
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Lecture Rules of Inference

4. Lecture 1.4 - Rules of Inference
Proofs – How do we know?
The following statements are true:
If I am Mila, then I am a great swimmer.
I am Mila.
What do we know to be true?
I am a great swimmer!
How do we know it?
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Lecture Rules of Inference

5. Proofs – How do we know?
A theorem is a statement that can be shown to be true.
A proof is the means of doing so.
Given set of true statements or previously proved theorems
Rules of inference
Proof
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Lecture Rules of Inference

6. Lecture 1.4 - Rules of Inference
What rules we study
Modus Ponens
Modus Tollens
Addition
Simplification
Disjunctive Syllogism
Hypothetical Syllogism
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Lecture Rules of Inference

7. Proofs – How do we know? The following statements are true:
If I have taken MA 106, then I am allowed to take CS 250
I have taken MA 106
What do we know to be true?
I am allowed to take CS 250
What rule of inference can we use to justify it?
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Lecture Rules of Inference

8. Rules of Inference – Modus Ponens
I have taken MA 106.
If I have taken MA 106, then I am allowed to take CS 250.
 I am allowed to take CS 250. p
p  q
 q
Inference Rule:
Modus Ponens
Tautology:
(p  (p  q))  q
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Lecture Rules of Inference

9. Rules of Inference – Modus Tollens
I am not allowed to take CS 250.
If I have taken MA 106 , then I am allowed to take CS 250.
 I have not taken MA 106. q
p  q
 p
Inference Rule:
Modus Tollens
Tautology:
(q  (p  q))  p
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Lecture Rules of Inference

10. Rules of Inference – Addition
I am not a great skater.
 I am not a great skater or I am tall. p
 p  q
Inference Rule:
Addition
Tautology:
p  (p  q)
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Lecture Rules of Inference

11. Rules of Inference – Simplification
I am not a great skater and you are sleepy.
 you are sleepy.
p  q
 p
Inference Rule:
Simplification
Tautology:
(p  q)  p
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Lecture Rules of Inference

12. Rules of Inference – Disjunctive Syllogism
I am a great eater or I am a great skater.
I am not a great skater.
 I am a great eater!
p  q q
 p
Inference Rule:
Disjunctive Syllogism
Tautology:
((p  q)  q)  p
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Lecture Rules of Inference

13. Rules of Inference – 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.
Inference Rule:
Hypothetical Syllogism
Tautology:
((p  q)  (q  r))  (p  r)
p  q
q  r
 p  r
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Lecture Rules of Inference

14. Lecture 1.4 - Rules of Inference
Examples
Addition
Amy is a computer science major.
 Amy is a math major or a computer science major.
If Ernie is a math major then Ernie is geeky.
Ernie is not geeky!
 Ernie is not a math major.
Modus Tollens
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Lecture Rules of Inference

15. Complex Example: Rules of Inference
Here’s what you know:
Ellen is a math major or a CS major.
If Ellen does not like discrete math, she is not a CS major.
If Ellen likes discrete math, she is smart.
Ellen is not a math major.
Can you conclude Ellen is smart?
M  C
D  C
D  S M
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Lecture Rules of Inference

16. Complex Example: Rules of Inference
1. M  C Given
2. D  C Given
3. D  S Given
4. M Given
5. C
DS (1,4)
6. D
MT (2,5)
7. S
MP (3,6)
Ellen is smart!
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Lecture Rules of Inference

17. Rules of Inference: Common Fallacies
Rules of inference, appropriately applied give valid arguments.
Mistakes in applying rules of inference are called fallacies.
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Lecture Rules of Inference

18. Rules of Inference: Common Fallacies
If I am Bonnie Blair, then I skate fast
I skate fast!
 I am Bonnie Blair
Nope
((p  q)  q)  p
Not a tautology.
If you don’t give me $10, I bite your ear.
I bite your ear!
 You didn’t give me $10.
Nope
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Lecture Rules of Inference

19. Rules of Inference: Common Fallacies
If it rains then it is cloudy.
It does not rain.
 It is not cloudy
Nope
((p  q)  p)  q
Not a tautology.
If it is a car, then it has 4 wheels.
It is not a car.
 It doesn’t have 4 wheels.
Nope
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Lecture Rules of Inference

20. Lecture 1.4 - Rules of Inference
Today’s Reading
Rosen 1.6
Please start solving the exercises at the end of each chapter section. They are fun.
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Lecture Rules of Inference