1. Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 11.2 and Notations

2. -Notation (Big O) Suppose f and g are two real-valued functions of a real variable x. If, for sufficiently large value of x, the value of | f | are less than those of a multiple of | g |, then f is of order at most g, or f ( x ) is O ( g ( x )). f is order at least g, written f ( x ) is O ( g ( x )), if and only if, there exist a positive real number B and a non negative real number b such that for all real number x > b. 2 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

3. -Notation (Big Omega) Suppose f and g are two real-valued functions of a real variable x. If, for sufficiently large value of x, the value of | f | are greater than those of a multiple of | g |, then f is of order at least g, or f ( x ) is ( g ( x )). f is order at least g, written f ( x ) is ( g ( x )), if and only if, there exist a positive real number B and a non negative real number b such that for all real number x > a. 3 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

4. -Notation (Big Theta) Suppose f and g are two real-valued functions of a real variable x. If, for sufficiently large value of x, the value of | f | are bounded both above and below by those of multiples of | g |, then f is of order g, or f ( x ) is ( g ( x )). f is order at least g, written f ( x ) is ( g ( x )), if and only if, there exist a positive real number B and a non negative real number k such that for all real number x > k. 4 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

5. Translating to Notations Use - notation to express the statement. for all real number x > 2. is. Use and O notations to express the statements. for all real numbers x > 0. is. for all real numbers x > 7. is. 5 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

6. Properties of Notations Let f and g be real-valued functions defined on the same set of nonnegative real numbers. 1.f ( x ) is Ω ( g ( x )) and f ( x ) is O ( g ( x )) if, and only if f ( x ) is Θ ( g ( x )). 2.f ( x ) is Ω ( g ( x )) if, and only if, g ( x ) is O ( f ( x )). 3.If f ( x ) is O ( g ( x )) and g ( x ) is O ( h ( x )), then f ( x ) is O ( h ( x )). 6 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

7. Order of Power Functions If 1 < x, then. In general, for any rational number r and s, if x > 1 and r < s, then. For any rational number r and s, If r < s, then is. 7 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

8. Polynomial Inequality Show that for any real number x, if x > 1, then. 8 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

9. Using the Definitions to Show That a Polynomial Function Coefficients Has a Certain Order Use the definitions of big-Omega, bit-O, and big-Theta to show that is. We must show that is. 9 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

10. A Big- O Approximation for a Polynomial with Some Negative Coefficients Use the definition of O -notation to show that is. Show that is for all integers s > 3. 10 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

11. A Big-Omega Approximation for a Polynomial with Some Negative Coefficients Use the definition of Ω -notation to show that is. Show that is for all integers r < 3. 11 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

12. Theorem 11.2.2 On Polynomial Orders Suppose are real numbers and. 1. is for all integers 2. is for all integers 3. is. Use the theorem on polynomial order to find orders for the functions given by the following formulas., for all real numbers x. What is order of the sum of the first n integers? 12 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

13. Theorem 11.2.3 Limitation on Orders of Polynomial Functions Let n be a positive integer, and let be real numbers with. If m is any integer with m < n, then is not and is not 13 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

14. Extension to Functions Composed of Rational Power Functions What is the order of 14 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University