1. Discrete Structures Prepositional Logic 2
Dr. Muhammad Humayoun
Assistant Professor
COMSATS Institute of Computer Science, Lahore.
Some of the material is taken from Dr. Muhammad Atif’s slides

2. Recap
Truth table: A truth table displays the relationship between the truth values of propositions. A table has 2 𝑛 rows where 𝑛 is number of proposition variables.
Exclusive or: ⊕ 𝑝⊕𝑞 is true when exactly one of 𝒑 and 𝒒 is true and is false otherwise.
Exercise: Draw a truth table of (𝑝⊕𝑞)∨(𝑝⇒𝑞)

3. Special Definitions 𝒑→𝒒 Inverse:¬𝒑→¬𝒒 Converse: 𝒒→𝒑
Contrapositive: ¬𝒒→¬𝒑

4. Example
Pakistani team wins whenever it is raining p: It is raining q: Pakistani team wins q whenever p ≡ if p, then q (𝑝→𝑞) If it is raining, then Pakistani team wins. Inverse:¬𝒑→¬𝒒 If it isn’t raining, then Pakistani team doesn’t win. Converse : 𝒒→𝒑 If Pakistani team wins, then it is raining. Contrapositive: ¬𝒒→¬𝒑 If Pakistani team doesn’t win, then it isn’t raining.

5. 𝑝 𝑞 𝑝 𝑞 𝑝  𝑞 𝑝  𝑞 𝑞  𝑝 𝑞  𝑝 𝑇 𝐹 Conditional Inverse Converse
Contrapositive 𝑝 𝑞 𝑝 𝑞
𝑝  𝑞
𝑝  𝑞
𝑞  𝑝
𝑞  𝑝 𝑇 𝐹

6. Conditional ≡Contrapositive
𝑝→𝑞 ≡¬𝑞→¬𝑝
Inverse ≡ Converse
¬𝑝→¬𝑞 ≡𝑞→𝑝

7. Biconditionals
Definition 6 Let p and q be propositions. The biconditional statement p ↔ q is the proposition “p if and only if q.” The biconditional statement p ↔ q is true when p and q have the same truth values, and is false otherwise. Biconditional statements are also called bi-implications.

8. Truth Table p ↔ q has exactly the same truth value as
(p → q) ∧ (q → p)

9. Common ways to express p ↔ q
“p is necessary and sufficient for q”
“if p then q, and conversely”
“p iff q”

10. Example
p: “You can take the flight” q: “You buy a ticket” p ↔ q: You can take the flight if and only if you buy a ticket You can take the flight iff you buy a ticket The fact that you can take the flight is necessary and sufficient for buying a ticket

11. You can take flight if and only if you buy a ticket
p: You can take flight
q: You buy a ticket
𝑝↔𝑞
You can take flight if and only if you buy a ticket
What is the truth value when:
you buy a ticket and you can take the flight ??
𝑇↔𝑇≡𝑇
you don’t buy a ticket and you can’t take the flight ??
𝐹↔𝐹≡𝑇
you buy a ticket but you can’t take the flight ??
𝑇↔𝐹≡𝐹
you can’t buy a ticket but can take the flight ??
𝐹↔𝑇≡𝐹

12. Precedence of Logical Operators
(𝑝⊕𝑞)∨(𝑝⇒𝑞)
Can be written as
(𝑝⊕𝑞)∨𝑝⇒𝑞
(T/F) ?
¬𝑎∧𝑏 𝑎∨𝑏⇔𝑏∨𝑎 𝑎∧𝑏∨𝑐
𝑎∧𝑏∧𝑐 𝑎∨𝑏∨𝑐

13. Exercise: For which values of a, b and c one gets 0 in the truth table of 𝑎∧ 𝑏⇒𝑐 ⇒( 𝑏⇒𝑎 ∧𝑐)

14. Logic and Bit Operations
Boolean values can be represented as 1 (true) and 0 (false)
A bit string is a series of Boolean values. Length of the string is the number of bits.
is eight Boolean values in one string
We can then do operations on these Boolean strings
Each column is its own boolean operation

15. 1.2 Applications of Propositional Logic
Translating English sentences (Formalization)
System Specifications
Boolean Searches
Logic circuits …

16. Translating English Sentences
You can access the Internet from campus only if you are a computer science major or you are not a freshman.
𝒂: You can access the Internet from campus
𝒄: You are a computer science major
𝒇: you are a freshman
𝒂→ (𝒄∨¬𝒇)

17. You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.
𝑟: you can ride roller coaster
𝑓 : you are under 4 feet
𝑜 : you are older than 16 years old
𝑓∧¬𝑜 →¬𝑟

18. System Specifications
The automated reply cannot be sent when the file system is full
p: The automated reply can be sent
q: The system is full
𝑞 ⟶¬𝑝

19. Consistency System specifications should be consistent,
They should not contain conflicting requirements that could be used to derive a contradiction
When specifications are not consistent, there would be no way to develop a system that satisfies all specifications

20. Determine whether these system specifications are consistent:
The diagnostic message is stored in the buffer or it is retransmitted.
The diagnostic message is not stored in the buffer.
If the diagnostic message is stored in the buffer, then it is retransmitted.

21. Determine whether these system specifications are consistent:
The diagnostic message is stored in the buffer or it is retransmitted.
The diagnostic message is not stored in the buffer.
If the diagnostic message is stored in the buffer, then it is retransmitted.
p: The diagnostic message is stored in the buffer
q: The diagnostic message is retransmitted
1. 𝒑∨𝒒 ¬𝒑 𝒑→𝒒

22. 1. 𝒑∨𝒒 ¬𝒑 𝒑→𝒒
Reasoning
An assignment of truth values that makes all three specifications true must have p false to make ￢𝑝 true.
Because we want 𝑝 ∨ 𝑞 to be true but 𝑝 must be false, q must be true.
Because 𝑝 → 𝑞 is true when 𝑝 is false and 𝑞 is true
we conclude that these specifications are consistent
Let us do it with truth table now

23. Is it remain consistent if the specification
“The diagnostic message is not retransmitted” is added?
p: The diagnostic message is stored in the buffer
q: The diagnostic message is retransmitted
1. 𝒑∨𝒒 ¬𝒑 𝒑→𝒒

24. Inconsistent Is it remain consistent if the specification
“The diagnostic message is not retransmitted” is added?
p: The diagnostic message is stored in the buffer
q: The diagnostic message is retransmitted
1. 𝒑∨𝒒 ¬𝒑 𝒑→𝒒
4. ¬𝒒
Inconsistent

25. Boolean Searches
Logical connectives are used extensively in searches of large collections of information, such as indexes of Web pages.
Because these searches employ techniques from propositional logic, they are called Boolean searches.

26. Finding Web pages about universities in New Mexico:
New AND Mexico AND Universities
‘New Mexico’ Universities
New Universities in Mexico
“New Mexico” AND Universities
(New AND Mexico OR Arizona) AND Universities
Arizona Universities
(Mexico AND Universities) NOT New

27. Quiz Let x = “لڑک” Then x + “ا” = لڑکا
Write Boolean search capturing this pattern

28. Logic Puzzles An island has two kinds of inhabitants,
Knights, who always tell the truth
Knaves, who always lie.
You encounter two people A and B.
What are A and B if
A says “B is a knight”
B says “The two of us are opposite types?

29. p: A is a knight ¬𝑝: A is a knave q: B is a knight ¬𝑞: B is a knave
A says “B is a knight”
B says “The two of us are opposite types?
p: A is a knight ¬𝑝: A is a knave
q: B is a knight ¬𝑞: B is a knave

30. p: A is a knight ¬𝑝: A is a knave q: B is a knight ¬𝑞: B is a knave
A says “B is a knight”
B says “The two of us are opposite types?
p: A is a knight ¬𝑝: A is a knave
q: B is a knight ¬𝑞: B is a knave
First possibility:
A is a knight; that is p is true.

31. p: A is a knight ¬𝑝: A is a knave q: B is a knight ¬𝑞: B is a knave
A says “B is a knight”
B says “The two of us are opposite types?
p: A is a knight ¬𝑝: A is a knave
q: B is a knight ¬𝑞: B is a knave
First possibility:
A is a knight; that is p is true.
If A is a knight, then he is telling the truth when he says that B is a knight, so that q is true, and A and B are the same type (both knight).
But, if B is a knight, then B’s statement that A and B are of opposite types (p ∧￢q) ∨ (￢p ∧ q), have to be true. But it is not; because A and B are both knights. Not consistent.
Conclusion: A is not a knight (p is false).

32. p: A is a knight ¬𝑝: A is a knave q: B is a knight ¬𝑞: B is a knave
A says “B is a knight”
B says “The two of us are opposite types?
p: A is a knight ¬𝑝: A is a knave
q: B is a knight ¬𝑞: B is a knave
Second possibility:
A is a knave; that is p is false.
If A is a knave, then he is telling lie when he says that B is a knight. So B is knave (q is false).
Also when B says that A and B are of opposite types (p ∧￢q) ∨ (￢p ∧ q), he again lies.
Conclusion: A and B are both knaves.

33. Logic Circuits
Propositional logic can be applied to the design of computer hardware
A logic circuit (or digital circuit) receives input signals 𝑝 1 , 𝑝 2 , , 𝑝 𝑛 , each a bit [either 0 (off) or 1 (on)], and produces output signals 𝑠 1 , 𝑠 2 , , 𝑠 𝑛 , each a bit.

34. Quiz: Draw 𝒑∧¬𝒒 ∨¬𝒓

35. Quiz: Draw 𝒑∧¬𝒒 ∨¬𝒓

36. 1.3 Propositional Equivalence
An important type of step used in a mathematical argument is the replacement of a statement with another statement with the same truth value
Propositional Equivalence is extensively used in the construction of mathematical arguments.

37. Tautology and Contradiction
A compound proposition which is always true, is called tautology. For example, ¬𝑝∨𝑝, 𝑎⇒𝑎, 𝑎⇒(𝑏⇒𝑎)
A compound proposition which is always false, is called contradiction. For example, ¬𝑝∧𝑝, ¬(𝑎⇒𝑎), 𝑎∧𝑏 ∧¬𝑎

38. Example on notebook:
𝑎⇒(𝑏⇒𝑎)
𝑎⇒𝑎

39. Logical Equivalences
Compound propositions that have the same truth values in all possible cases are called logically equivalent.
The compound propositions p and q are called logically equivalent if p ↔ q is a tautology.
The notation p ≡ q denotes that p and q are logically equivalent.

40. Show that ¬ 𝑝∨𝑞 ≡¬𝑝∧¬𝑞

41. Standard equivalences
Identity
𝑝 ∧ 𝑻 ≡ 𝑝
𝑝 ∨ 𝑭 ≡ 𝑝
Domination
𝑝 ∨ 𝑻 ≡ 𝑻
𝑝 ∧ 𝑭 ≡ 𝑭

42. Standard equivalences
Idempotence
𝑝∧𝑝≡𝑝
𝑝∨𝑝≡𝑝
Double Negation
¬¬𝑝≡𝑝

43. Standard Equivalences
Commutative law:
𝑝∧𝑞≡𝑞∧𝑝
𝑝∨𝑞≡𝑞∨𝑝
𝑝⇔𝑞≡𝑞⇔𝑝

44. Standard equivalences
Associativity
𝑝∧𝑞 ∧𝑟≡𝑝∧ 𝑞∧𝑟
𝑝∨𝑞 ∨𝑟≡𝑝∨ 𝑞∨𝑟
𝑝⇔𝑞 ⇔𝑟≡𝑝⇔(𝑞⇔𝑟)

45. Standard equivalences
Inversion ¬𝑇≡𝐹 ¬𝐹≡𝑇
Negation ¬𝑝 ≡ (𝑝⇒𝐹)
Contradiction 𝑝∧¬𝑝 ≡ 𝐹

46. Distributive Law
𝑝∧ 𝑞∨𝑟 ≡ 𝑝∧𝑞 ∨ 𝑝∧𝑟
𝑝∨ 𝑞∧𝑟 ≡ 𝑝∨𝑞 ∧ 𝑝∨𝑟

48. ¬ 𝑝∧𝑞 ≡¬𝑝∨¬𝑞 ¬ 𝑝∨𝑞 ≡¬𝑝∧¬𝑞 De Morgan’s Law
¬( 𝑝 1 ∧ 𝑝 2 ∧ · · · ∧ 𝑝 𝑛 ) ≡ (¬ 𝑝 1 ∨¬ 𝑝 2 ∨ ··· ∨¬ 𝑝 𝑛 )
¬ 𝑝∨𝑞 ≡¬𝑝∧¬𝑞
¬ ( 𝑝 1 ∨ 𝑝 2 ∨ · · · ∨ 𝑝 𝑛 )≡ (¬ 𝑝 1 ∧¬ 𝑝 2 ∧ ··· ∧ ¬𝑝 𝑛 )

49. Generalization De Morgan’s Laws ¬ 𝑖=1 𝑛 𝑝 𝑖 ≡ 𝑖=1 𝑛 ¬ 𝑝 𝑖
𝑖=1 𝑛 𝑝 𝑖 𝑐𝑎𝑛 𝑏𝑒 𝑢𝑠𝑒𝑑 𝑓𝑜𝑟 𝑝 1 ∧ 𝑝 2 ∧…∧ 𝑝 𝑛
𝑖=1 𝑛 𝑝 𝑖 𝑐𝑎𝑛 𝑏𝑒 𝑢𝑠𝑒𝑑 𝑓𝑜𝑟 𝑝 1 ∨ 𝑝 2 ∨…∨ 𝑝 𝑛
De Morgan’s Laws
¬ 𝑖=1 𝑛 𝑝 𝑖 ≡ 𝑖=1 𝑛 ¬ 𝑝 𝑖
¬( 𝑖=1 𝑛 𝑝 𝑖 ) ≡ 𝑖=1 𝑛 ¬𝑝 𝑖

51. Absorption laws
𝑝 ∨ (𝑝 ∧ 𝑞) ≡ 𝑝
𝑝 ∧ 𝑝 ∨ 𝑞 ≡ 𝑝

52. Negation laws
𝑝 ∨￢𝑝 ≡ 𝑻
𝑝 ∧￢𝑝 ≡ 𝑭

53. Implication
𝑝⇒𝑞≡¬𝑝∨𝑞
𝑝∨𝑞≡¬𝑝⇒𝑞

54. More Implication Laws 𝑝 → 𝑞 ≡ ￢𝑞 →￢𝑝 𝑝 ∧ 𝑞 ≡ ￢(𝑝 →￢𝑞) ¬(𝑝 → 𝑞) ≡ 𝑝 ∧￢𝑞
(𝑝 → 𝑞) ∧ (𝑝 → 𝑟) ≡ 𝑝 → (𝑞 ∧ 𝑟)
(𝑝 → 𝑟) ∧ (𝑞 → 𝑟) ≡ (𝑝 ∨ 𝑞) → 𝑟
(𝑝 → 𝑞) ∨ (𝑝 → 𝑟) ≡ 𝑝 → (𝑞 ∨ 𝑟)
(𝑝 → 𝑟) ∨ (𝑞 → 𝑟) ≡ (𝑝 ∧ 𝑞) → 𝑟

55. Bi-implications 𝑝 ↔ 𝑞 ≡ (𝑝 → 𝑞) ∧ (𝑞 → 𝑝) 𝑝 ↔ 𝑞 ≡ ￢𝑝 ↔￢𝑞
𝑝 ↔ 𝑞 ≡ (𝑝 ∧ 𝑞) ∨ (￢𝑝 ∧￢𝑞)
￢(𝑝 ↔ 𝑞) ≡ 𝑝 ↔￢𝑞

56. Using Logical Equivalence
Show that ￢(𝑝 → 𝑞) and 𝑝 ∧￢𝑞 are logically equivalent.
Show that ￢(𝑝 ∨ (￢𝑝 ∧ 𝑞)) and ￢𝑝 ∧￢𝑞 are logically equivalent by developing a series of logical equivalences.
Prove that (𝑝∧𝑞)⇒(𝑝∨𝑞) is a tautology.

57. Using Logical Equivalence
Ex: Prove that 𝑝∧𝑞 ⇒(𝑝∨𝑞) is a tautology. To show that this statement is a tautology, we will use logical equivalences to demonstrate that it is logically equivalent to T 𝑝 ∧ 𝑞 → 𝑝 ∨ 𝑞 ≡¬ 𝑝 ∧ 𝑞 ∨ 𝑝 ∨ 𝑞 Implication equivalence ≡ ¬𝑝∨¬𝑞 ∨ 𝑝 ∨ 𝑞 1st De Morgan law ≡¬𝑝∨(¬𝑞∨ 𝑝 ∨ 𝑞 ) Associative law ≡¬𝑝∨(𝑝∨ ¬q ∨ 𝑞 ) Commutative law ≡(¬𝑝∨𝑝)∨ ¬q ∨ 𝑞 Associative law ≡𝑇∨𝑇 Tautologies ≡𝑇 Idempotence

58. Do Exercises