1. Discrete Structures for Computer Science
Presented By: Andrew F. Conn
Slides adapted from: Adam J. Lee
Lecture #2: Logic Puzzles and Propositional Equivalence
August 31st, 2016

2. Announcements
The course website is up Homework 1 will be posted to the website today Due: Before class, two Wednesday’s from now 9/14 Monday’s class is cancelled Your schedule reflects this

3. Today’s topics Some more English translation. Logic puzzles.
Propositional equivalences.

4. English sentences can often be translated into propositional sentences
But why would we do that?!?
Reasoning about law
Philosophy and epistemology
Verifying complex system specifications
Note on Aristotle, mechanization of law, system specifications, RFCs

5. Find logical connectives Create logical expression
Example #1
Example: You can see an R-rated movie only if you are over 17 or you are accompanied by your legal guardian.
Let:
r  “You can see an R-rated movie”
o  “You are over 17”
a  “You are accompanied by your legal guardian
Translation: r  (o  a)
Find logical connectives
Translate fragments
Create logical expression

6. Example #2
Example: You can have free coffee if you are senior citizen and it is a Tuesday
Let:
c  “You can have free coffee”
s  “You are a senior citizen”
t  “It is a Tuesday”
Translation: (s  t)  c

7. Example #3
Example: If you are under 17 and are not accompanied by your legal guardian, then you cannot see the R-rated movie.
Let:
r  “You can see the R-rated movie”
u  “You are under 17”
a  “You are accompanied by your legal guardian
Translation: (u  ¬a)  ¬r
Note: The above translation is the contrapositive of the translation from example 1!

8. A technical support conundrum
Alice and Bob are technical support agents. If an agent is having a bad day, he or she will always lie to you. If an agent is having a good day, he or she will always tell you the truth. Alice tells you that Bob is having a bad day. Bob tells you that he and Alice are both having the same type of day. Can you trust the advice you receive from Alice during your call?
How do we solve this
type of problem?

9. Solving logic puzzles is easy!
Step 1: Identify rules and constraints
Step 2: Assign propositions to key concepts
Step 3: Make assumptions and reason logically!

10. Technical support revisited
Alice and Bob are technical support agents. If an agent is having a bad day, he or she will always lie to you. If an agent is having a good day, he or she will always tell you the truth. Alice tells you that Bob is having a bad day. Bob tells you that he and Alice are both having the same type of day. Can you trust the advice you receive from Alice during your call?
Step 1: Identify the rules of the puzzle
Good day = tell the truth
Bad day = lie!
Step 2: Assign propositions to the key concepts in the puzzle
a  “Alice is having a good day”
b  “Bob is having a good day”

11. Step 3: Working it out
If Alice is having a good day, then she is telling the truth about Bob and he is a liar. This is consistent with his statement that Alice and he are having the same type of day.
If Alice is having a bad day, then she is telling a lie about Bob and he is having a good day. Thus Bob always tells the truth and he and Alice are having the same day. This implies that Alice is having a good day. This contradicts the assumption that we made about Alice having a bad day.
Formal Proof: The second proof presented in symbols!
¬𝒂 →𝒃
𝒃→𝒂∧𝒃
¬𝒂∧𝒂 ⊥∎

12. Another example
Consider a group of friends: Frank, Anna, and Chris. If Frank is not the oldest, then Anna is. If Anna is not the youngest, then Chris is the oldest. Determine the relative ages of Frank, Anna, and Chris.
Propositions:
𝑓≡ “Frank is the oldest”
𝑎≡ “Anna is the oldest”
𝑎 ′ ≡ “Anna is the youngest”
𝑐≡ “Chris is the oldest”
Rules:
¬f→𝑎
¬ 𝑎 ′ →𝑐

13. Step 3: Working it out
Argument 1: Assume Frank is not the oldest. Then Anna is the oldest based on Rule 1. Then Chris is the oldest based on Rule 2. This is a contradiction and thus we know Frank is the oldest.
Argument 2: Assume Anna is not the youngest. Then Chris is the oldest by Rule 2. This contradicts with Frank being the oldest and thus Anna is the youngest.
By process of elimination we have that Chris is the middle child.

14. Sometimes no solution is a solution!
Alice and Bob are technical support agents. Alice says “I am having a good day.” Bob says “I am having a good day.” Can you trust either Alice or Bob?
Step 1: Identify rules
Good day = tell the truth
Bad day = lie
Step 2: Assign propositions
a = Alice is having a good day
b = Bob is having a good day

15. Group work!
Problem 1: Alice and Bob are technical support agents working to fix your computer. Alice tells you that Bob is having a bad day today and that you should expect a long wait before your computer is fixed. Bob tells you not to worry, Alice is just having a bad day---your computer will be ready in no time.
Question: Can you draw any conclusions about when your computer will be fixed? If so, what
can you learn?

16. Another word on Implication
There are some important definitions related to implication
Given p  q:
q  p is its converse
¬q  ¬p is its contrapositive
¬p  ¬q is its inverse
The contrapositive of the converse!
Are any of these equivalent to p  q?
Yes, implication and its contrapositive always have the same truth value.
Why might this
be useful?
Why: Maybe easier to prove…

17. Propositional equivalences: preliminaries
Definition: A tautology is a compound proposition that is always true, regardless of the truth values of the propositions occurring within it.
Definition: A contradiction is a compound proposition that is always false, regardless of the truth values of the propositions occurring within it.
Definition: A contingency is a compound proposition whose truth value is dependent on the propositions occurring within it.

18. Examples
Are the following compound propositions tautologies, contradictions, or contingencies? p ¬p
p  ¬p T F
p  ¬p
¬p  p
p  q
tautology p ¬p
p  ¬p T F
contradiction
contingency
Draw truth tables on the board p q
p  q T F

19. What are logical equivalences and why are they useful?
Definition: Compound propositions p and q are logically equivalent if p  q is a tautology. The notation p  q means that p and q are logically equivalent.
Logical equivalences are extremely useful!
Aid in the construction of proofs
Allow us to simplify compound propositions
Example: r  s  r  s
Let: p  r  s
q  r  s
This will help many of you recognize when implications are true! r s p q
p  q T F

20. DeMorgan’s laws allow us to distribute negation over compound propositions
If “p or q” isn’t true, then neither p nor q is true
Two laws:
(p  q)  p  q
(p  q)  p  q
Prove: (p  q)  p  q
If “p and q” isn’t true, then at least one of p or q is false p q p q
p  q
(p  q)
p  q T F

21. End of 8/31/16 Lecture
We ran a little too close on time and needed to stop here.
The rest of this slide deck will be reviewed on 9/7/16

22. Using DeMorgan’s laws
Use DeMorgan’s laws to negate the following expressions:
“Bob is wearing blue pants and a sweatshirt”
b  s
(b  s)  b  s
Bob is not wearing blue pants or is not wearing a sweatshirt
“I will drive or I will walk”
d  w
(d  w)  d  w
I will not drive and I will not walk

23. Group work!
Problem 2: Prove that ¬(p  q) and ¬p  ¬q are logically equivalent, i.e., ¬(p  q)  ¬p  ¬q. This is the second DeMorgan’s law.
Problem 3: Use DeMorgan’s laws to negate the following propositions:
Today I will go running or ride my bike
Tom likes both pizza and beer

24. Sometimes using truth tables to prove logical equivalencies can become cumbersome
Recall that for an equivalence with n propositions, we need to build a truth table with 2n rows
Fine for tables with n = 2, 3, or 4
Consider n = 30---we would need 1,073,741,824 rows in the truth table!
Another option: Direct manipulation of compound propositions using known logical equivalencies

25. There are many useful logical equivalences
Name
p  T  p
p  F  p
Identity laws
p  F  F
p  T  T
Domination laws
p  p  p
p  p  p
Idempotent laws
(p)  p
Double negation law
p  q  q  p
p  q  q  p
Commutative laws
(p  q)  p  q

26. More useful logical equivalences
Name
(p  q)  r  p  (q  r)
(p  q)  r  p  (q  r)
Associative laws
p  (q  r)  (p  q)  (p  r)
p  (q  r)  (p  q)  (p  r)
Distributive laws
(p  q)  p  q
¬(p  q)  ¬p  ¬q
DeMorgan’s laws
p  (p  q)  p
p  (p  q)  p
Absorption laws
p  p  T
p  p  F
Negation laws
More equivalencies in the book!

27. Prove: (p  q)  (p  r)  p  (q  r)

28. Prove that (p  q)  (p  q) is a tautology

29. Final Thoughts
Logic can help us solve real world problems and play challenging games
Logical equivalences help us simplify complex propositions and construct proofs
More on proofs later in the course
Next time:
Predicate logic and quantification
Please read section 1.3

30. Extra slides

31. Technique #2: Truth table
The previous argument can actually be captured in a truth table
Step 1: Identify the rules of the puzzle
Good day = tell the truth
Bad day = lie!
Step 2: Assign propositions to the key concepts in the puzzle
a  “Alice is having a good day”
b  “Bob is having a good day”
Step 3: Assign propositions to the claims made by Alice and Bob
ca  “Bob is having a bad day”
cb  “Alice and I are having the same type of day”

32. Step 4: Fill in the truth table
This is the only row that is consistent with the rules of the puzzle! a b ca cb T F
Possible good day/bad day combinations
Truth values of claims ca and cb

33. Formal argumentation (cont.)
Step 3 (cont.):
Assume that Alice is having a good day (i.e., proposition a is true)
Since Alice is telling the truth, we know that Bob is having a bad day (i.e., ¬b)
Since Bob is lying, his claim that he and Alice are having the same type of day, is actually stating that he and Alice are having different types of day.
Therefore, the assumption that Alice is having a good day is consistent with the rules of the puzzle.
Result: Alice is having a good day and we can trust her for all of our tech support needs!
Step 3: Make assumptions and reason logically about these propositions
Assume Alice is having a bad day and is thus lying (i.e., ¬a).
Since she says Bob is having a bad day, this means that Bob is actually having a good day (i.e., b).
Bob’s claim that he and Alice are having the same type of day is a contradiction, because Alice is having a bad day (by assumption) and Bob cannot lie (since proposition b is true).
This contradiction tells us that Alice cannot be having a bad day.

34. Solution Assume that Anna is the oldest Assume that Anna is the middle
Contradiction: If Anna isn’t the youngest, Chris must be the oldest (by rule 2), and we can’t have two oldest people.
Thus, Frank is the oldest (by rule 1)
Assume that Anna is the middle
Contradiction: If Anna isn’t the youngest, Chris must be the oldest (by rule 2), but rule 1 tells us that Frank is the oldest.
Solution: Frank is the oldest, Chris is in the middle, and Anna is the youngest.

35. can use this equivalence
Example derivation
We proved this
earlier in lecture!
Prove: (p  q)  (p  r)  p  (q  r)
(p  q)  (p  r)  (¬p  q)  (¬p  r)
 (¬p  ¬p)  (q  r)
 ¬p  (q  r)
 p  (q  r)
a  b  a  b (twice)
Commutative and associative laws
Idempotent law
a  b  a  b
Equivalence is
bidirectional, so we
can use this equivalence
both ways