Fall 2014 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 7.3 Composition of Functions

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of Two Functions Let and be functions with the property that the range of f is a subset of the domain of g and Define a new function as follows: where is read g circle f and is read g of f of x. The function is called the composition of f and g. 2

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of Functions Defined by Formulas Suppose two functions and defined as below: a. Find the compositions and. b. Is ? Explain. 3

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of Functions Defined by Formulas Suppose two functions and defined as below: a. Find the compositions and. b. Is ? Explain. 4

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of Functions Defined by Formulas Suppose two functions and defined as below: a. Find the compositions and. b. Is ? Explain. 5

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of Functions Defined on Finite Sets Let and. Define two functions and by the arrow diagram below. Draw the arrow diagram for. What is the range of ? 6 fg Y’

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of Functions Defined on Finite Sets – cont’ Let and. Define two functions and by the arrow diagram below. Draw the arrow diagram for. What is the range of ? Range is { y, z } 7 Fall 2010 COMP 4605/5605 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University fg Y’

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition with the Identity Function If f is a function from a set X to a set Y, and is the identity function on X, and is the identity function on Y, then 8 Roughly, this is because More formal proof is given in the textbook.

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composing a Function with Its Inverse If is a one-to-one and onto function with inverse function then 9 Roughly, this is because More formal proof is given in the textbook.

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of One-to-One Functions Suppose we have two one-to-one functions f and g. Is their composite function one-to-one? 10

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of One-to-One Functions Suppose we have two one-to-one functions f and g. Is their composite function one-to-one? Yes! Proof Suppose we have we two different elements such that Since we have Also, since f is one-to-one has to be true. Since g is also one-to one, has to be true. (Contradiction!) 11

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of Onto Functions Suppose we have two onto functions and. Is their composite function still onto? 12

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of Onto Functions Suppose we have two onto functions and. Is their composite function still onto? Yes 13

Fall 2014 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Composition of Onto Functions Suppose we have two onto functions and. Is their composite function still onto? Yes Proof Since g is onto, there has to be such that. Also, since f is onto, there has to be some such that. Therefore, it is true. 14