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Lecture Discrete structures: Chapter 10 - Amer Rasheed

This chapter includes contents: Uses a explicit linear ordering, insertions and removals are performed individually, there are no restrictions on objects inserted into (pushed onto) the queue - that object is designated the back of the queue,. This topic discusses the concept of a queue: Description of an Abstract Queue, list applications, implementation, queuing theory, standard template library. | (CSC 102) Lecture 10 Discrete Structures Previous Lectures Summary Divisors Prime Numbers Fundamental Theorem of Arithmetic Division Algorithm . Greatest common divisors. Least Common Multiple Relative Prime Elementary Number Theory II Today's Lecture Rational Number Properties of Rational Numbers Irrational numbers Absolute values Triangular inequality Floor and Ceiling functions Rational Numbers A real number r is rational if, and only if, r = a/b for some integers a and b with b ≠ 0. A real number that is not rational is irrational. More formally, r is a rational number ↔ ∃ integers a and b such that r = a/b, b ≠ 0. Determine whether following numbers are rational? 10/3 -(5/39) 2/0 0.121212121212 Cont . Yes, 10/3 is quotient of the integers 10 and 3. Yes, -5/39, which is a quotient of the integers. No, 2/0 is not a number (division by 0 is not allowed) Yes, Let x=0.1212121212121, then 100x=12.12121212 . 100x – x = 12.12121212 - 0.1212121212 = 12. 99∙x = 12 and so x=12/99. Therefore 0.12121212 = 12/99, which is a ratio of two non zero integers and thus is a rational number. Properties Of Rational Number Theorem: Every Integer is a rational Number Proof: Its obvious by the definition of rational Number. Properties Of Rational Number Theorem: The sum of any two rational Numbers is rational. Proof: Suppose r and s are rational numbers. Then by definition of rational, r = a/b and s = c/d for some integers a, b, c, and d with b ≠ 0 and d ≠ 0. So r + s = a/b + c/d = (ad + bc)/b∙d. Let p = ad + bc and q = bd. Then p and q are integers because product and sum of integers are integers. Also q ≠ 0. Thus r+s = p/q. so r+s is a rational number. Properties Of Rational Number Theorem: The double of a rational number is rational. Proof: Suppose r is rational number. Then 2r = r + r is a sum of two rational number and is a rational number. Irrational Numbers The number which is not rational is called irrational number. i.e., a decimal representation which is neither . | (CSC 102) Lecture 10 Discrete Structures Previous Lectures Summary Divisors Prime Numbers Fundamental Theorem of Arithmetic Division Algorithm . Greatest common divisors. Least Common Multiple Relative Prime Elementary Number Theory II Today's Lecture Rational Number Properties of Rational Numbers Irrational numbers Absolute values Triangular inequality Floor and Ceiling functions Rational Numbers A real number r is rational if, and only if, r = a/b for some integers a and b with b ≠ 0. A real number that is not rational is irrational. More formally, r is a rational number ↔ ∃ integers a and b such that r = a/b, b ≠ 0. Determine whether following numbers are rational? 10/3 -(5/39) 2/0 0.121212121212 Cont . Yes, 10/3 is quotient of the integers 10 and 3. Yes, -5/39, which is a quotient of the integers. No, 2/0 is not a number (division by 0 is not allowed) Yes, Let x=0.1212121212121, then 100x=12.12121212 . 100x – x = 12.12121212 - 0.1212121212 = 12. 99∙x = 12 and so x=12/99. .