LECTURE 1

Disrete mathematics and its application by rosen 7 th edition THE FOUNDATIONS: LOGIC AND PROOFS 1.1 PROPOSITIONAL LOGIC

 A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both  = 2 (true)  = 13 (true)  Islamabad is capital of Pakistan (true)  Karachi is the largest city of Pakistan (true)  = 111 (false)  Some sentences are not prepositions  Where is my class? (un decelerated sentence)  What is the time by your watch? (un decelerated sentence)  x + y = ? ( will be prepositions when value is assigned)  Z +w * r = p PROPOSITIONS

 We use letters to denote propositional variables (or statement variables).  The truth value of a proposition is true, denoted by T, if it is a true proposition.  The truth value of a proposition is false, denoted by F, if it is a false proposition.  Many mathematical statements are constructed by combining one or more propositions. They are called compound propositions, are formed from existing propositions using logical operators. PROPOSITIONS

 Definition: Let p be a proposition. The negation of p, denoted by ￢ p (also denoted by p), is the statement “It is not the case that p.”  The proposition ￢ p is read “not p.” The truth value of the negation of p, ￢ p, is the opposite of the truth value of p.  Also denoted as “ ′ ”  Examples:  p := Sir PC is running Windows OS  ￢ p := sir PC is not running Windows OS  p := a + b = c  p := a + b ≠ c NEGATION The Truth Table for the Negation of a Proposition p ￢p￢p TFTF FTFT

 Definition: Let p and q be propositions. The conjunction of p and q, denoted by p ∧ q, is the proposition “p and q.” The conjunction p ∧ q is true when both p and q are true and is false otherwise.  Also known as UNION, AND, BIT WISE AND, AGREGATION  Denoted as ^, &, AND CONJUNCTION The Truth Table for the conjunction of a Proposition p qp ^ q T T F F T F TFFFTFFF

 Definition: Let p and q be propositions. The disjunction of p and q, denoted by p ∨ q, is the proposition “p or q.” The disjunction p ∨ q is false when both p and q are false and is true otherwise.  Also known as OR, BIT WISE OR, SEGREGATION  Denoted as v, ||, OR DISJUNCTION The Truth Table for the conjunction of a Proposition p qp v q T T F F T F TTTFTTTF

 Definition: Let p and q be propositions. The exclusive or of p and q, denoted by p ⊕ q, is the proposition that is true when exactly one of p and q is true and is false otherwise.  Also known as ZORING  Denoted as XOR, Ex OR, ⊕ EXCLUSIVE OR The Truth Table for the conjunction of a Proposition p q p ⊕ q T T F F T F FTTFFTTF

 Let p and q be propositions. The conditional statement p → q is the proposition “if p, then q.” The conditional statement p → q is false when p is true and q is false, and true otherwise.  In the conditional statement p → q, p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence).  Denoted by  CONDITIONAL STATEMENT The Truth Table for the conjunction of a Proposition p qp → q T T F F T F TFTTTFTT

 The proposition q → p is called the converse of p → q.  The converse, q → p, has no same truth value as p → q for all cases.  Formed from conditional statement. CONVERSE

 The contrapositive of p → q is the proposition ￢ q → ￢ p.  only the contrapositive always has the same truth value as p → q.  The contrapositive is false only when ￢ p is false and ￢ q is true.  Formed from conditional statement. CONTRAPOSITIVE The Truth Table for the CONTRAPOSITIVE of a Proposition p q ￢ p ￢ q ￢ p → ￢ q T T F F T F F T F T T F F F T T TTFTTTFT

 Formed from conditional statement.  The proposition ￢ p → ￢ q is called the inverse of p → q.  The converse, q → p, has no same truth value as p → q for all cases. INVERSE

 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. BICONDITIONAL The Truth Table for the CONTRAPOSITIVE of a Proposition q pq ↔ p T T F F T F TFFTTFFT

 Definition: When more that one above defined preposition logic combines it is called as compound preposition.  Example:  (p^q)v(p’)  (p ⊕ q) ^ (r v s) COMPOUND PROPOSITIONS

 (p ∨￢ q) → (p ∧ q) COMPOUND PROPOSITIONS (TRUTH TABLE) The Truth Table of (p ∨￢ q) → (p ∧ q) pq ￢q￢qp ∨￢ qp ∧ q(p ∨￢ q) → (p ∧ q) TTFFTTFF TFTFTFTF FTFTFTFT TTFTTTFT TFFFTFFF TFTFTFTF

Precedence of Logical Operators. OperatorPrecedence ￢ 1 ^2 v3 →4 ↔5 XOR6 PRECEDENCE OF LOGICAL OPERATORS

 Computers represent information using bits  A bit is a symbol with two possible values, namely, 0 (zero) and 1 (one).  A bit can be used to represent a truth value, because there are two truth values, namely, true and false.  1 bit to represent true and a 0 bit to represent false. That is, 1 represents T (true), 0 represents F (false).  A variable is called a Boolean variable if its value is either true or false. Consequently, a Boolean variable can be represented using a bit. LOGIC AND BIT OPERATIONS Truth ValueBit T1 F0

 Computer bit operations correspond to the logical connectives.  By replacing true by a one and false by a zero in the truth tables for the operators ∧ (AND), ∨ (OR), and ⊕ (XOR), the tables shown for the corresponding bit operations are obtained. LOGIC AND BIT OPERATIONS Table for the Bit Operators OR, AND, and XOR. pqp ^ qp v qp XOR q

 bitwise OR bitwise AND bitwise XOR  bitwise OR bitwise AND bitwise XOR BITWISE OR, BITWISE AND, AND BITWISE XOR