2. Chapter 1
Logic and Proofs
Homework 1
Two merchants publish new marketing campaign to attract more customers. The first merchant launches a motto “Good stuffs are not cheap.” The second merchants says “Cheap stuffs is not good.”
Examine whether both mottos tell the same message or not.
In your opinion, which motto is better?
Solution:
p: “Good stuffs.”
q: “Cheap stuffs.”
The motto of the first merchant: p  q.
The motto of the second merchant: q  p
Is it true that p  q  q  p

3. Solution to Homework 1 p q p q p  q q  p T F  p  q  q  p
Chapter 1
Logic and Proofs
Solution to Homework 1 p q p q
p  q
q  p T F
 p  q  q  p
The mottos of both merchants are logically equivalent (thus identical).

4. Variants of Conditional Proposition
Chapter 1
Logic and Proofs
Variants of Conditional Proposition
For an implication given by p  q, we can create:
The converse, given by q  p
The inverse, given by p  q
The contrapositive, given by q  p p q p q
Implication
p  q
Converse
q  p
Inverse
p  q
Contra-positive
q  p T F

5. Variants of Conditional Proposition
Chapter 1
Logic and Proofs
Variants of Conditional Proposition
Example:
“If it snows, then the traffic moves slowly.”
p: “It snows.”
q: “The traffic moves slowly.”
The converse “If the traffic moves slowly, then it snows.”
The inverse “If it does not snow, then the traffic does not move slowly.”
The contrapositive “It the traffic does not move slowly, then it does not snow.”
Converse: q  p
Inverse: p  q
Contrapositive: q  p

6. Variants of Conditional Proposition
Chapter 1
Logic and Proofs
Variants of Conditional Proposition
Example:
Determine the converse, inverse, and contrapositive of: “If someone owns a car, then she is rich.”
The converse “If someone is rich, then she owns a car.”
The inverse “If someone does not own a car, then she is not rich.”
The contrapositive “If someone is not rich, then she does not own a car.”
Converse: q  p
Inverse: p  q
Contrapositive: q  p

7. Chapter 1
Logic and Proofs
Contrapositive
Example: Determine the contrapositive of the following propositions.
“If she is found guilty, then she will be put in prison.”
“If 6 is greater than 0, then 6 is not a negative number.”
“Ivan passes the exam only if he studies hard.” ·p only if q
Solution:
“If she is not put in prison, then she was not found guilty.”
“If 6 is a negative number, then 6 is not greater than 0.”
“If Ivan does not study hard, then he does not pass the exam.”
Contrapositive: q  p

8. Chapter 1
Logic and Proofs
Contrapositive
Example: Determine the contrapositive of the following propositions.
“Only if she is not late, she will get the job.” ·p only if q
“Wind is necessary for the kite to fly.” ·q is necessary for p
“Rain falling in the afternoon is sufficient to make the night freezing cold.” ·p is sufficient for q
Solution:
“If she is late, then she will not get the job.”
“If the wind does not blow, then the kite cannot fly.”
“If the night is not freezing cold, then rain does not fall in the afternoon.”
Contrapositive: q  p

9. Biconditional Proposition
Chapter 1
Logic and Proofs
Biconditional Proposition
Biconditional proposition is also called bi-implication.
Notation: p  q
The way to pronounce: “p if and only if q”
Biconditional p q
p  q T F
The biconditional p  q is true when p and q have the same truth values and is false otherwise.

10. Biconditional Proposition
Chapter 1
Logic and Proofs
Biconditional Proposition
Various ways to express implication p  q:
p if and only if q.
p is the necessary and sufficient condition for q.
p is necessary and sufficient for q.
if p then q, and conversely (vise versa).
if p then q and if q then p.
p iff q.

11. Biconditional Proposition
Chapter 1
Logic and Proofs
Biconditional Proposition p q
p  q
p  q
q  p
(p  q)  (q  p) T F
p  q  (p  q)  (q  p)
“p if and only if q”  “if p then q and if q then p”

12. Biconditional Proposition
Chapter 1
Logic and Proofs
Biconditional Proposition
Example: Biconditional in various expressions.
“1 + 1 = 2 if and only if = 4.”
“The necessary and sufficient condition for rain to fall is high air humidity.”
“If you are rich, then you have much money, and vice versa.”
“Cikarang is located in West Java iff West Java is one of the 34 provinces in Indonesia.”
“If the weather is hot then I buy ice cream, and if I buy ice cream then the weather is hot.”
“Training and discipline are sufficient and necessary for you to win the match.”
“You are promoted if you have connection, and you have connection if you are promoted.”
“If I watch television too much then my eyes are irritated, and vice versa.”
“Trains always come late when and only when I need them for travel.”

13. Logical Equivalences for Conditional and Biconditional
Chapter 1
Logic and Proofs
Logical Equivalences for Conditional and Biconditional

14. Chapter 1
Logic and Proofs
Some Examples
Example: Given a proposition “It is not true that he learns Technical Drawing but not State Philosophy.”,
Express this proposition in symbolic notation (logical expression).
Write one possible logically equivalent proposition as the proposition above.
Solution:
Defining p: “He learns Technical Drawing.” q: “He learns State Philosophy.”
(p  q)
(p  q)  p  q “He does not learn Technical Drawing or learns State Philosophy.” p q
(p  q)
p  q T F

15. Some Examples Example: Construct a truth table for (p  q)  (p  q)
Chapter 1
Logic and Proofs
Some Examples
Example: Construct a truth table for (p  q)  (p  q)

16. Chapter 1
Logic and Proofs
Some Examples
Example: Three propositions are given to describe the quality of a hotel.
p: “The service is good.”
q: “The room rate is low.”
r: “The hotel is a three-star hotel.”
Translate the following proposition into symbolic notation:
“The room rate is low but the service is bad.”
“Either the room rate is high or the service is good, but not both.”
“It is not the case that if a hotel is a three star hotel, then the room rate is low and the service is bad.”
q  p
q  p
(q  p )  (q  p)
(r  (q  p))

17. Chapter 1
Logic and Proofs
Some Examples
Example: Express the following statement in symbolic notation: “If you are below 17 years old, then you may not vote in a general election, unless you are already married.”
Solution:
Defining p: “You are below 17 years old.” q: “You are already married.” r: “You may vote in a general election.”
Then the statement above can be express in symbolic notation as: (p  q)  r, “If you are below 17 years old and are not already married, then you may not vote in a general election.”
which is equivalent to r  (p  q)

18. Some Examples Example: Proof that [p  (p  q)]  q is a tautology.
Chapter 1
Logic and Proofs
Some Examples
Example: Proof that [p  (p  q)]  q is a tautology.
Solution:
To proof the tautology, we construct the truth table: p q p
p  q
p  (p  q)
[p  (p  q)]  q T F
 [p  (p  q)]  q is a tautology.
Redo the proof by using logical equivalences.

19. Chapter 1
Logic and Proofs
Argument and Proof
The truth value of some statements about the world is obvious and easy to assign.
The truth of other statements may not be so obvious, but it may still follow (be derived) from known facts about the world.
To show the truth value (validity) of such a statement following from other statements, we need to provide a correct supporting argument or proof.
Important questions:
When is the argument correct?
How to proof that a statement is correct?
How to construct a correct argument, what method to use?

20. Chapter 1
Logic and Proofs
Argument and Proof
A formal proof is a set of proofs which follows logically from the set of premises.
Formal proofs allow us to infer new true statements from known true statements.
A proposition or its part can be transformed using a sequence of logical equivalence until some conclusions can be reached.
Example: Show that (p  q)  p is a tautology by using logical equivalences.

21. Chapter 1
Logic and Proofs
Argument and Proof
In showing the truth of a statement, we must use a valid argument.
A valid argument is one built using a so called rules of inference from premises. When all premises are true, then the argument should lead us to a the correct conclusion.
Typically, a problem of finding a valid argument looks like this:
(p1  p2  p3  …  pn)  q
Premises
Conclusion
Conclusion
Premises

22. Rules of Inference p  q p  q p  q q  p
Chapter 1
Logic and Proofs
Rules of Inference
An inference rule is a logical form consisting of a function which takes premises, analyzes their syntax (grammatical arrangement), and returns a conclusion or conclusions.
The inference rule is valid in the sense that if the premises are true, then so is the conclusion.
Popular rules of inference in propositional logic include modus ponens and modus tollens.
Modus ponens takes two premises, one in the form “if p then q” and another in the form “p”, and returns the conclusion “q”.
Modus tollens takes two premises, one in the form “if p then q” and another in the form “q”, and returns the conclusion “p”.
p  q p
 q
p  q q
 p
Modus Ponens
Modus Tollens

23. Rules of Inference p  q p  q p  q q  p
Chapter 1
Logic and Proofs
Rules of Inference
1. Modus ponens Given the implication p  q is true and p is true, then q is true. p q
p  q
p  (p  q) T F
p  q p
 q
Modus Ponens
2. Modus tollens Given the implication p  q is true and q is true, then p is true. p q
p  q
q  (p  q) F T
p  q q
 p
Modus Tollens

24. Rules of Inference p  q p  q p  q q  p
Chapter 1
Logic and Proofs
Rules of Inference
Modus ponens “If today is Tuesday, then John will go to work.” “Today is Tuesday.” “Therefore, John will go to work.”
“If a low tide occurs after an underwater earthquake, then a tsunami is triggered.” “A low tide occurs after an underwater earthquake.” “Therefore, a tsunami is triggered.”
p  q p
 q
Modus Ponens
Modus tollens “If the dog detects an intruder, then the dog will bark. “ “The dog did not bark.” “Therefore, no intruder was detected by the dog.”
p  q q
 p
Modus Tollens

25. Checking the Validity of an Argument
Chapter 1
Logic and Proofs
Checking the Validity of an Argument
The validity of an argument can be check by using truth table.
If [p1  p2  ….  pn ]  q is a tautology, then the argument is valid.

26. Checking the Validity of an Argument
Chapter 1
Logic and Proofs
Checking the Validity of an Argument
Example: Show that the reasoning of the following argument is false, or the argument is invalid: “If a low tide occurs after an underwater earthquake, then a tsunami is triggered.” “A tsunami is triggered.” “Therefore, a low tide occurs after an underwater earthquake.”
Solution: Taking p : “A low tide occurs after an underwater earthquake.” q : “A tsunami is triggered.”
then the argument above can be written as:
p  q (p1)
q (p1)
 p (q) p1 p2 q
[p1  p2 ]  q
p q p
[(p q)  q]  p T F
[(p q)  q]  p is not a tautology.
The argument is not valid.

27. Rules of Inference p  p  q p  q  p p  q  q
Chapter 1
Logic and Proofs
Rules of Inference
3. Addition Given p is true, then p  q will always be true. p
 p  q
Addition
4. Simplification Given p  q is true, then p (and q) will always be true.
p  q
 p
Simplification
p  q
 q
Simplification

28. Example: Checking the Argument
Chapter 1
Logic and Proofs
Example: Checking the Argument
Example: We have the following premises: “It is not sunny this afternoon and it is colder than yesterday.” “We will go swimming only if it is sunny.” “If we do not go swimming, then we will take a canoe trip.” “If we take a canoe trip, then we will be home by sunset.”
Can we conclude that “Therefore, we will be home by sunset”?
Solution: Defining p : “It is sunny this afternoon.” q : “It is colder than yesterday.” r : “We will go swimming.” s : “We will take a canoe trip.” t : “We will be home by sunset.”
then the argument above can be written as:
p  q
r  p
r  s
s  t
 t
We must now check whether this argument is valid or not

29. Example: Checking the Argument
Chapter 1
Logic and Proofs
Example: Checking the Argument
p  q
 p
1st premise
p  q
r  p
r  s
s  t
 t
Simplification p
r  p
 r
2nd premise
Modus tollens r
r  s
 s
3rd premise
Modus ponens s
s  t
 t
4th premise
Modus ponens
The argument is VALID

30. More Rules of Inference
Chapter 1
Logic and Proofs
More Rules of Inference
5. Conjunction If p and q are each true, then p  q is true. p q
 p  q
Conjunction
6. Disjunctive syllogism If p  q is true and p is false, then q must be true.
p  q p
 q
Disjunctive syllogism
7. Resolution If p  q is true and p  r is true, then q  r must be true.
p  q
p  r
 q  r
Resolution
8. Hypothetical syllogism If p  q is true, and q  r is true, then p  r must be true.
p  q
q  r
 p  r
Hypothetical syllogism

31. Summary: Rules of Inference
Chapter 1
Logic and Proofs
Summary: Rules of Inference
Modus ponens p
p  q
 q
Modus tollens q
 p
Addition
 p  q
Simplification
p  q
Conjunction q
 p  q
Disjunctive syllogism
p  q p
Resolution
p  r
 q  r
Hypothetical syllogism
q  r
 p  r

32. More Example: Checking the Argument
Chapter 1
Logic and Proofs
More Example: Checking the Argument
Example: Given the following premises: “If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on.” “If the sailing race is held, then the trophy will be awarded.” “The trophy was not awarded.”
Proof that the conclusion “Therefore, it rained” is valid.
Solution: Defining r : “It is rain.” f : “It is foggy.” s : “The sailing race will be held.” d : “The lifesaving demonstration will go on.” t : “The trophy will be awarded .”
then the argument above can be written as:
(r  f)  (s  d)
s  t t
 r
We must now check whether this argument is valid or not

33. More Example: Checking the Argument
Chapter 1
Logic and Proofs
More Example: Checking the Argument
(s  d)  (r  f)
(A) Contrapositive of 1st premise
(s  d)  (r  f)
(B) De Morgan’s Law and Double Negation Law of (A)
s  t t
 s
(r  f)  (s  d)
s  t t
 r
2nd premise
3rd premise
Modus tollens s
 s  d
(C) Addition
(s  d)  (r  f)
s  d
 (r  f)
Modus ponens of (B) and (C)
(r  f)
 r
Simplification
The argument is VALID

34. Chapter 1
Logic and Proofs
Homework 2
Given the statement “A valid password is necessary for you to log on to the campus server.”
Express the statement above in the proposition form of “if p then q.”
Determine also the negation, converse, inverse, and contrapositive of the statement.
Check the validity of the argument below by using truth table: “If 5 is less then 4, then 5 is not a prime number.” “5 is not less then 4.” “Therefore, 5 is a prime number.”

35. Chapter 1
Logic and Proofs
Homework 2A
Given the statement “In order to be registered for consultation with our doctors, you only have to send an SMS to the number on this card.”
Express the statement above in the proposition form of “if p then q.”
Determine also the negation, converse, inverse, and contrapositive of the statement.
Check the validity of the argument below by using truth table and rules of inference: “The budget is proposed and the venue is already booked.” “If the deadline is reduced then the budget will not be proposed.” “Therefore, the deadline is not reduced.”