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## Discrete Structures Prepositional Logic 2

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**Discrete StructuresPrepositional Logic 2**Dr. Muhammad Humayoun Assistant 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**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**Special Definitions**Inverse: Converse: Contrapositive:**Example**Pakistani team wins whenever it is raining p: It is raining q: Pakistani team wins q whenever pif 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.**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. Biconditionalstatements are also called bi-implications.**Truth Table**• p ↔ qhas exactly the same truth value as (p → q) ∧ (q → p)**Common ways to express p ↔ q**• “p is necessary and sufficient for q” • “if p then q, and conversely” • “p iffq”**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 iffyou buy a ticket The fact that you can take the flight is necessary and sufficient for buying a ticket**p: You can take flight**q: You buy a ticket 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 ?? • 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 ??**Precedence of Logical Operators**Can be written as (T/F) ?**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 • Each column is its ownbooleanoperation**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. You can access the Internet from campus You are a computer science major 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. you can ride roller coaster you are under 4 feet 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 would be no way to develop a system that satisfies all specifications**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.**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. 2. 3.**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**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.**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**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.**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) NOTNew**Quiz**• Let x = “لڑک” Then x + “ا” = لڑکا Write Boolean search capturing this pattern**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?**A says “B is a knight”**• B says “The two of us are opposite types? p: A is a knight : A is a knave q: B is a knight : B is a knave**A says “B is a knight”**• B says “The two of us are opposite types? p: A is a knight : A is a knave q: B is a knight : B is a knave First possibility: A is a knight; that is p is true.**A says “B is a knight”**• B says “The two of us are opposite types? p: A is a knight : A is a knave q: B is a knight : 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).**A says “B is a knight”**• B says “The two of us are opposite types? p: A is a knight : A is a knave q: B is a knight : 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.**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.**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.**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, , ,**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.**Standard equivalences**Identity Domination**Standard equivalences**Idempotence Double Negation**Standard Equivalences**Commutative law:**Standard equivalences**Associativity**Standard equivalences**• Inversion • Negation • Contradiction**Generalization**De Morgan’s Laws