Discrete Structures
Prepositional Logic 2
Dr. Muhammad Humayoun
Assistant Professor
Recap
• Truth table:
A truth table displays the relationship between
the truth values of propositions. A table has 2𝑛
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:
Special Definitions
𝒑 → 𝒒
Inverse:¬𝒑 → ¬𝒒
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
Conditional Inverse Converse Contrapositive
𝑝 𝑞 𝑝 𝑞 𝑝 𝑞 𝑝 𝑞 𝑞 𝑝 𝑞 𝑝
𝑇 𝑇 𝐹 𝐹 𝑇 𝑇 𝑇 𝑇
𝑇 𝐹 𝐹 𝑇 𝐹 𝑇 𝑇 𝐹
•
Conditional ≡ Contrapositive
•
𝑝 → 𝑞 ≡ ¬𝑞 → ¬𝑝
•
Inverse
≡
Converse
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
Truth Table
• p ↔ q has exactly the same truth value as
Common ways to express
p
↔
q
• “p is necessary and sufficient for q”
• “if p then q, and conversely”
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
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 ??
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
Exercise:
For which values of a, b and c one
gets 0 in the truth table of
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.
– 10110100 is eight Boolean values in one string
• We can then do operations on these Boolean
strings
1.2 Applications of Propositional Logic
• Translating English sentences (Formalization)
• System Specifications
• Boolean Searches
• Logic circuits
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
• 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
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
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
Determine whether these system specifications are
consistent:
1. The diagnostic message is stored in the buffer or it is retransmitted.
2. The diagnostic message is not stored in the buffer.
Determine whether these system specifications are
consistent:
1. The diagnostic message is stored in the buffer or it is retransmitted.
2. The diagnostic message is not stored in the buffer.
3. If the diagnostic message is stored in the buffer, then it is retransmitted.
1. 𝒑 ∨ 𝒒 2. ¬𝒑 3. 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
• 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
• 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 2. ¬𝒑 3. 𝒑 → 𝒒 4. ¬𝒒
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
• 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
– ‘New Mexico’ Universities
– Arizona Universities
Quiz
• Let x = “کڑل”
Then x + “ا” = اکڑل
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”
– 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
– 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 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
– 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.
Logic Circuits
• Propositional logic can be applied to the design
of computer hardware
• A logic circuit (or digital circuit) receives input
signals 𝑝1, 𝑝2, . . . , 𝑝𝑛, each a bit [either 0 (off) or
1 (on)], and produces output signals
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
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,
Example on notebook:
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.
Standard equivalences
Identity
•
𝑝 ∧ 𝑻 ≡ 𝑝
•
𝑝 ∨ 𝑭 ≡ 𝑝
Domination
Standard equivalences
Idempotence
•
𝑝 ∧ 𝑝 ≡ 𝑝
•
𝑝 ∨ 𝑝 ≡ 𝑝
Double Negation
Standard Equivalences
Commutative law:
•
𝑝 ∧ 𝑞 ≡ 𝑞 ∧ 𝑝
•
𝑝 ∨ 𝑞 ≡ 𝑞 ∨ 𝑝
Standard equivalences
Associativity
•
𝑝 ∧ 𝑞 ∧ 𝑟 ≡ 𝑝 ∧ 𝑞 ∧ 𝑟
•
𝑝 ∨ 𝑞 ∨ 𝑟 ≡ 𝑝 ∨ 𝑞 ∨ 𝑟
Standard equivalences
• Inversion
¬𝑇 ≡ 𝐹 ¬𝐹 ≡ 𝑇
• Negation
¬𝑝 ≡ (𝑝 ⇒ 𝐹)
Distributive Law
•
𝑝 ∧ 𝑞 ∨ 𝑟 ≡ 𝑝 ∧ 𝑞 ∨ 𝑝 ∧ 𝑟
De Morgan’s Law
•
¬ 𝑝 ∧ 𝑞 ≡ ¬𝑝 ∨ ¬𝑞
• ¬(𝑝1 ∧ 𝑝2 ∧ · · · ∧ 𝑝𝑛) ≡ (¬𝑝1 ∨ ¬𝑝2 ∨ ··· ∨ ¬𝑝𝑛)
•
¬ 𝑝 ∨ 𝑞 ≡ ¬𝑝 ∧ ¬𝑞
Generalization
• 𝑛𝑖=1 𝑝𝑖 𝑐𝑎𝑛 𝑏𝑒 𝑢𝑠𝑒𝑑 𝑓𝑜𝑟 𝑝1 ∧ 𝑝2 ∧ ⋯ ∧ 𝑝𝑛
• 𝑛𝑖=1 𝑝𝑖 𝑐𝑎𝑛 𝑏𝑒 𝑢𝑠𝑒𝑑 𝑓𝑜𝑟 𝑝1 ∨ 𝑝2 ∨ ⋯ ∨ 𝑝𝑛
De Morgan’s Laws
•
¬
𝑛
𝑖=1
𝑝
𝑖
≡
𝑛
𝑖=1
¬𝑝
𝑖
Absorption laws
•
𝑝 ∨ (𝑝 ∧ 𝑞) ≡ 𝑝
Negation laws
•
𝑝 ∨
￢
𝑝 ≡ 𝑻
Implication
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)
Bi-implications
• 𝑝 ↔ 𝑞 ≡ (𝑝 → 𝑞) ∧ (𝑞 → 𝑝)
• 𝑝 ↔ 𝑞 ≡ ￢𝑝 ↔ ￢𝑞
• 𝑝 ↔ 𝑞 ≡ (𝑝 ∧ 𝑞) ∨ (￢𝑝 ∧ ￢𝑞)
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.
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
