tailieunhanh - Lecture Discrete structures: Chapter 21 - Amer Rasheed

In this chapter, the following content will be discussed: Inverse functions, finding an inverse function, composition of functions, composition of functions, composition of functions defined on finite sets, plotting functions. | (CSC 102) Lecture 21 Discrete Structures Previous Lecture Summery Sum/Difference of Two Functions Equality of Two Functions One-to-One Function Onto Function Bijective Function (One-to-One correspondence) Today’s Lecture Inverse Functions Finding an Inverse Function Composition of Functions Composition of Functions Composition of Functions defined on finite sets Plotting Functions Inverse Functions Theorem The function F-1 is called inverse function. Inverse Functions Given an arrow diagram for a function. Draw the arrow diagram for the inverse of this function Finding an Inverse Function The function f : R → R defined by the formula f (x) = 4x − 1, for all real numbers x Theorem Composition of Functions Composition of Functions defined on finite sets Let X = {1, 2, 3}, Y ’ = {a, b, c, d}, Y = {a, b, c, d, e}, and Z = {x, y, z}. Define functions f : X → Y’ and g: Y → Z by the arrow diagrams below. Draw the arrow diagram for g ◦ f . What is the range of g ◦ f ? Composition of Functions defined on finite sets To find the arrow diagram for g ◦ f , just trace the arrows all the way across from X to Z through Y . The result is shown below. Composition of Functions defined on finite sets Let X = {1, 2, 3}, Y ’ = {a, b, c, d}, Y = {a, b, c, d, e}, and Z = {x, y, z}. The range of g ◦ f is {y, z}. Composition of Functions defined on Infinite Sets Let f : Z → Z, and g: Z → Z be two functions. ., f (n)=n + 1 for all n ∈ Z and g(n) = n2 for all n ∈ Z. a. Find the compositions g◦f and f◦g. b. Is g ◦ f = f ◦g? Explain. The functions g ◦ f and f ◦g are defined as follows: (g ◦ f )(n) = g( f (n)) = g(n + 1) = (n + 1)2 for all n ∈ Z, ( f ◦g)(n) = f (g(n)) = f (n2) = n2 + 1 for all n ∈ Z. Composition of Functions defined on Infinite Sets Let f : Z → Z, and g: Z → Z be two functions. ., f (n)=n + 1 for all n ∈ Z and g(n) = n2 for all n ∈ Z. b. Is g ◦ f = f ◦g? Explain. Composition with Identity Function Let X = {a, b, c, d} and Y = {u, v,w}, and suppose f : X → Y is given by . | (CSC 102) Lecture 21 Discrete Structures Previous Lecture Summery Sum/Difference of Two Functions Equality of Two Functions One-to-One Function Onto Function Bijective Function (One-to-One correspondence) Today’s Lecture Inverse Functions Finding an Inverse Function Composition of Functions Composition of Functions Composition of Functions defined on finite sets Plotting Functions Inverse Functions Theorem The function F-1 is called inverse function. Inverse Functions Given an arrow diagram for a function. Draw the arrow diagram for the inverse of this function Finding an Inverse Function The function f : R → R defined by the formula f (x) = 4x − 1, for all real numbers x Theorem Composition of Functions Composition of Functions defined on finite sets Let X = {1, 2, 3}, Y ’ = {a, b, c, d}, Y = {a, b, c, d, e}, and Z = {x, y, z}. Define functions f : X → Y’ and g: Y → Z by the arrow diagrams below. Draw the arrow diagram for g ◦ f . What is the range of g ◦ f ? Composition of Functions

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• Discrete structures
• Lecture Discrete structures
• Cấu trúc rời rạc
• Discrete data
• Discrete mathematics
• Computer scientists
• Equivalence relations