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Discrete StructuresPrepositional Logic 2

Dr. Muhammad HumayounAssistant Professor

COMSATS Institute of Computer Science, Lahore.mhumayoun@ciitlahore.edu.pk

https://sites.google.com/a/ciitlahore.edu.pk/dstruct/

Some of the material is taken from Dr. Muhammad Atif’s slides

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Recap• Truth table:

A truth table displays the relationship between the truth values of propositions. A table has rows where is number of proposition variables.

• Exclusive or: is true when exactly one of and is true and is false otherwise.

• Exercise: Draw a truth table of

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Special Definitions

Inverse:

Converse:

Contrapositive:

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ExamplePakistani team wins whenever it is rainingp: It is rainingq: Pakistani team winsq whenever p if p, then q If it is raining, then Pakistani team wins.Inverse:If it isn’t raining, then Pakistani team doesn’t win.Converse : If Pakistani team wins, then it is raining.Contrapositive: If Pakistani team doesn’t win, then it isn’t raining.

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Conditional Inverse Converse Contrapositive

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• Inverse Converse

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Biconditionals

Definition 6Let p and q be propositions. The biconditional statement p ↔ q is the proposition “p ifand 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.

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Truth Table

• p ↔ q has exactly the same truth value as (p → q) (q → p)∧

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Common ways to express p ↔ q

• “p is necessary and sufficient for q”• “if p then q, and conversely”• “p iff q”

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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

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p: You can take flightq: You buy a ticket

You can take flight if and only if you buy a ticketWhat is the truth value when:• you buy a ticket and you can take the flight ??

• you don’t buy a ticket and you can’t take the flight ??

• you buy a ticket but you can’t take the flight ??

• you can’t buy a ticket but can take the flight ??

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Precedence of Logical Operators

Can be written as

(T/F) ?

¬𝑎∧𝑏 𝑎∧𝑏∧𝑐

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Exercise:For which values of a, b and c one gets 0 in the truth table of

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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– Each column is its own

boolean operation

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1.2 Applications of Propositional Logic

• Translating English sentences (Formalization)• System Specifications• Boolean Searches • Logic circuits• …

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Translating English Sentences

• You can access the Internet from campus only if you are a computer science major or you are not a freshman.

You can access the Internet from campus You are a computer science major you are a freshman

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• You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.

you can ride roller coaster you are under 4 feetyou are older than 16 years old

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System Specifications

• The automated reply cannot be sent when the file system is fullp: The automated reply can be sentq: The system is full

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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

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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.

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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.p: The diagnostic message is stored in the buffer q: The diagnostic message is retransmitted 1. 2. 3.

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1. 2. 3. Reasoning• An assignment of truth values that makes all three

specifications true must have p false to make true.

• Because we want to be true but must be false, q must be true.

• Because is true when is false and is true• we conclude that these specifications are

consistent• Let us do it with truth table now

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• 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. 2. 3.

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• 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. 2. 3. 4.

Inconsistent

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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.

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• 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

• (Mexico AND Universities) NOT New

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Quiz

• Let x = “لڑک”Then x + “ لڑکا” = اWrite Boolean search capturing this pattern

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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?

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– A says “B is a knight” – B says “The two of us are opposite types?

p: A is a knight : A is a knaveq: B is a knight : B is a knave

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– A says “B is a knight” – B says “The two of us are opposite types?

p: A is a knight : A is a knaveq: B is a knight : B is a knaveFirst possibility:A is a knight; that is p is true.

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– A says “B is a knight” – B says “The two of us are opposite types?

p: A is a knight : A is a knaveq: B is a knight : B is a knaveFirst 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).

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– A says “B is a knight” – B says “The two of us are opposite types?

p: A is a knight : A is a knaveq: B is a knight : B is a knaveSecond 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.

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Logic Circuits

• Propositional logic can be applied to the design of computer hardware

• A logic circuit (or digital circuit) receives input signals , each a bit [either 0 (off) or 1 (on)], and produces output signals , each a bit.

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Quiz: Draw

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Quiz: Draw

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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.

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Tautology and Contradiction

• A compound proposition which is always true, is called tautology. For example, , ,

• A compound proposition which is always false, is called contradiction. For example, , ,

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Example on notebook:

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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.

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• Show that

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Standard equivalences

Identity

Domination

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Standard equivalences

Idempotence

Double Negation

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Standard Equivalences

Commutative law:

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Standard equivalences

Associativity

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Standard equivalences• Inversion

• Negation

• Contradiction

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Distributive Law

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De Morgan’s Law

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Generalization

De Morgan’s Laws

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Absorption laws

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Negation laws

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Implication

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More Implication Laws

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Bi-implications

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Using Logical Equivalence

• Show that and are logically equivalent.

• Show that and are logically equivalent by developing a series of logical equivalences.

• Prove that is a tautology.

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Using Logical EquivalenceEx: Prove that 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

Implication equivalence 1st De Morgan law

Tautologies Idempotence

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Do Exercises