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Discrete Structures Chapter 1 Part B Fundamentals of Logic Nurul Amelina Nasharuddin Multimedia Department 1 Predicates • A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables • The domain of a predicate variable is a set of all values that may be substituted in place of the variable • P(x): x is a student at UPM • P(x, y): x is a student at y 2 Predicates • The sets in which predicate variables take their values may be described either in words or in symbols • x A indicates that x is an element of the set A • x A; x is not an element of the set A • {1, 2, 3} refers to the set whose elements are 1, 2 and 3 • Certain set of numbers that frequently referred to are Symbol Set Example R Set of all real numbers All points on a line number Z Set of all integers …,-3, -2, -1, 0, 1, 2, 3, … Q Set of all rational numbers, or quotient {p/q | p ∈ Z, q ∈ Z, q ≠ 0} of integers 3 Predicates • If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements in D that make P(x) true when substituted for x. The truth set is denoted as: {x D | P(x)} which is read “the set of all x in D such that P(x)” • True or false?: Let P(x): x2 > x with domain set R (real numbers). 4 Predicates • Let Q(n) be the predicate “n is a factor of 8.” Find the truth set of Q(n) if – the domain of n is the set of Z+ – the domain of n is the set of Z • Ans: •{1, 2, 4, 8} •{-1, -2, -4, -8, 1, 2, 4, 8} 5 Predicates • Let P(x) and Q(x) be predicates with the common domain D • P(x) Q(x) means that every element in the truth set of P(x) is in the truth set of Q(x), or equivalently for every x, P(x) Q(x) • P(x) Q(x) means that P(x) and Q(x) have identical truth sets, or equivalently for every x, P(x) Q(x) 6 Universal Quantifier, • Let P(x) be a predicate with domain D. • A universal statement is a statement in the form “x D, P(x)”. denotes “for all” • It is true iff P(x) is true for every x from D. It is false iff P(x) is false for at least one x from D. • A value of x for which P(x) is false is called a counterexample to the universal statement • Examples – D = {1, 2, 3, 4, 5}: x D, x² >= x (True) – x R, x² >= x (False) • Method of exhaustion 7 Existential Quantifier, • Let P(x) be a predicate with domain D. • An existential statement is a statement in the form “x D, P(x)”. denotes “there exists” • It is true iff P(x) is true for at least one x from D. It is false iff P(x) is false for every x from D. • Examples: – m Z, m² = m (True) – E = {5, 6, 7, 8, 9}, x E, m² = m (False) 8 Quantifier • Rewrite the following without using and : a. x R, x2 0 b. x Z such that m2 = m • Ans: a. All real numbers have nonnegative squares The square of any real number is nonnegative b. m2 =m, for some integer m Some integers equal their own square 9 Universal Conditional Statement • Universal conditional statement “x, if P(x) then Q(x)”: • Eg: x R, if x > 2, then x2 > 4 • Universal conditional statement is called vacuously true or true by default iff P(x) is false for every x in D • Eg: When x =1, if 1 > 2, then 12 > 4 (True) When x = -3, if (-3) > 2, then (-3)2 > 4 (True) 10 Negation of Quantified Statements • The negation of a universally quantified statement x D, P(x) is x D, ~P(x) (x D, P(x)) x D such that ~P(x) • The negation of an existentially quantified statement x D, P(x) is x D, ~P(x) ( x D, P(x)) x D such that ~P(x) • Rewrite formally, formal and informal negation: No politicians are honest 11 Answer No politicians are honest • Formal version: politicians x, x is not honest • Formal negation: a politician x such that x is honest • Informal negation: Some politicians are honest 12 Negation of Quantified Statements • The negation of a universal conditional statement x D, P(x) Q(x) is x D, P(x) ~Q(x) (x, if P(x) then Q(x)) x such that P(x) and ~Q(x) • Noted before; ~(p q) p ~q • Write formal negation: people p, of p is blond then p has blue eyes • Ans: a person such that p is blond and p does not have blue eyes 13 Exercises • Write negations for each of the following statements: – All dinosaurs are extinct – No irrational numbers are integers – Some exercises have answers – All COBOL programs have at least 20 lines – The sum of any two even integers is even – The square of any even integer is even 14 Multiply Quantified Statements • Imagine you are in a factory and the guide tell you “There is person supervising every detail of the production process” • Have both existential quantifier There is and universal quantifier every • When a statement contains more than one quantifier, the actions being performed is in the order in which the quantifier occur. 15 Multiply Quantified Statements • To establish the truth of the statement “x in D, y in E such P(x, y)” Pick whatever x from set D , then find element y in E that “works” for that particular y in P(x, y) • To establish the truth of the statement “x in D, such that y in E, P(x, y)” Find element x in D that “works” no matter y in P(x, y) 16 Multiply Quantified Statements • For all positive numbers x, there exists number y such that y < x • There exists number x such that for all positive numbers y, y < x • For all people x there exists person y such that x loves y (Informally: Everybody loves somebody) • There exists person x such that for all people y, x loves y (Informally: Somebody loves everybody) 17 Negation of Multiply Quantified Statements • The negation of x, y, P(x, y) is logically equivalent to x, y, ~P(x, y) • The negation of x, y, P(x, y) is logically equivalent to x, y, ~P(x, y) 18 Negation of Multiply Quantified Statements • Rewrite formally and negate: Everybody trusts somebody. • Formally: (a) people x, a person y such that x trusts y. • Negation: Somebody trusts nobody. • Formally: (b) a person x, such that people y, x does not trust y. 19 Necessary and Sufficient Conditions, Only If • x, r(x) is a sufficient condition for s(x) means: x, if r(x) then s(x) • x, r(x) is a necessary condition for s(x) means: x, if s(x) then r(x) • x, r(x) only if s(x) means: x, if r(x) then s(x) 20 Logical Equivalence and Logical Implication for Quantified Statements in One Variable x D, (P(x) Q(x) (x D, P(x)) (x D, Q(x)) x D, (P(x) Q(x)) (x D, P(x)) (x D, Q(x)) (x D, P(x)) (x D, Q(x)) x D, (P(x) Q(x)) x D, (P(x) Q(x)) (x D, P(x)) (x D, Q(x)) 21 Application: Digital Logic Circuits • Digital Logic Circuit is a basic electronic component of a digital system • Values of digital signals are 0 or 1 (bits) • Black Box is specified by the signal input/output table • Input/output table = truth table • Three gates: AND-gate, OR-gate, NOT-gate Application: Digital Logic Circuits • Combinational circuit is a combination of logical gates • Combinational circuit always correspond to some Boolean expression, such that input/output table of a table and a truth table of the expression are identical 23 Application: Digital Logic Circuits • Finding Boolean expression for a circuit: P ~ (Q R) 24 Application: Digital Logic Circuits • Determining output for a given input: Move from left to right through the diagram, tracing the action of each gate on the input signals • Constructing the input/output table for a circuit: list the all possible combinations for input signals, and find the output for each by tracing through the unit 25 Application: Digital Logic Circuits • A recognizer is a circuit that outputs 1 for exactly one particular combination of input signals and outputs 0’s for all other combinations • Multiple-input AND and OR gates • Finding a circuit that corresponds to a given input/output table: – Construct equivalent Boolean expression using disjunctive normal form: for all outputs of 1 construct a conjunctive form based on the truth table row. All conjunctive forms are united using disjunction – Construct a digital logic circuit equivalent to the Boolean expression Application: Digital Logic Circuits • Given • Boolean expression: (P Q R) Input Output P Q R S 1 1 1 1 (P Q R) 1 1 0 0 (P Q R) 1 0 1 1 1 0 0 1 • Circuit: 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 27 Application: Digital Logic Circuits • Two digital logic circuits are equivalent iff their input/output tables are identical • Simplification of circuits by using Theorem 1.1.1 to simplify the Boolean expression • Another way to simplify circuit by using – Scheffer stroke, | (NAND, AND-gate followed by NOT-gate) – Peirce arrow, (NOR, OR-gate followed by NOT-gate) Exercises • Construct circuits for (P Q) R • From the given input/output Input Output P Q R S table below, construct 1 1 1 0 (a) a Boolean expression 1 1 0 1 1 0 1 0 (b) a circuit 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 0