1. Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 7.2 One-to-One and Onto, Inverse Functions

2. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University One-to-One Functions Let F be a function from a set X to a set Y. F is one-to- one (or injective) if, and only if, for all elements and in X, or equivalently, Symbolically, 2

3. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Examples 3 Not one-to-one One-to-one

4. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University One-to-One Functions on Infinite Sets Now suppose f is a function defined on an infinite set X. By definition, f is one-to-one if, and only if, the following universal statement is true: Thus, to prove f is one-to-one, you suppose and are elements of X such that and need to show To show that f is not one-to-one, you need to show a counter example in which there are two elements and in X so that but 4

5. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Example Define: and by the rules and Q) Is f one-to-one? Prove or give a counterexample. 5

6. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Example – cont’ Define: and by the rules and Q) Is f one-to-one? Prove or give a counterexample. A) If not, there have to be two and such that and However, and this is a contradiction! 6

7. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Example – cont’ Define: and by the rules and Q) Is g one-to-one? Prove or give a counterexample. 7

8. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Example – cont’ Define: and by the rules and Q) Is g one-to-one? Prove or give a counterexample. A) 8

9. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Onto Functions Let F be a function from a set X to a set Y. F is onto (or subjective) if, and only if, given any element y in Y, it is possible to find an element x in X with the property that y = F ( x ). Symbolically: 9

10. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Examples 10 Not one-to-one Not onto One-to-one and onto Not one-to-one but onto

11. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Onto Functions on Infinite Sets Now suppose f is a function defined on an infinite set X. By definition, f is one-to-one if, and only if, the following universal statement is true: Thus, to prove f is onto, you suppose that is any element of Y and need to show there is an element x of X with To show that f is not onto, you need to show a counter example in which there is an elements y of Y such that for any x in X. 11

12. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Example Define: and by the rules and Q) Is f onto? Prove or give a counterexample. A) For any f(x), we have 12

13. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Example – cont’ Define: and by the rules and Q) Is h onto? Prove or give a counterexample. A) Let h(n) = 6. Then, there is no n such that 13

14. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Relations between Exponential and Logarithmic Functions For positive number, the exponential function with base b, denoted, is the function from R to defined as follows: For all real number x, where and. 14

15. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Law of Exponents If b and c are any positive real numbers and u and v are any real numbers, the following laws of exponents hold true: 15

16. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Logarithmic Function The logarithmic function with base b was defined for any positive number to be the function from to R with the property that for each positive real number x, the exponent to which b must be raised to obtain x. 16

17. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Properties of Logarithms For any positive real numbers b, c and x with and : 17

18. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University One-to-One Correspondences A one-to-one correspondence (or bijection) from a set X to a set Y is a function that is both one- to-one and onto. 18 Not one-to-one Not onto One-to-one and onto Not one-to-one but onto

19. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Inverse Function Suppose is a one-to-one correspondence; that is, suppose F is one-to-one and onto. Then there is a function that is defined as follows: Given any element y in Y, = that unique element x in X such that F ( x ) equals y. In other words, Such is called the inverse function for F. 19

20. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Inverse Function 20 Not one-to-one Not onto One-to-one and onto Not one-to-one but onto

21. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Inverse Function 21

22. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Inverse Function 22 Not a function

23. Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Inverse Function Theorem 7.2.3 If X and Y are two sets and is one-to-one and onto, then is also one-to-one and onto. Example 7.2.13 23