1. 03 May 2009Instructor: Tasneem Darwish1 University of Palestine Faculty of Applied Engineering and Urban Planning Software Engineering Department Introduction to Discrete Mathematics Functions Part2

2. 03 May 2009Instructor: Tasneem Darwish2 Outlines Composite Functions Injections and Surjections

3. 03 May 2009Instructor: Tasneem Darwish3 Let f : A → B and g : B → C be functions, then the composite of f and g is the function from A to C g ◦ f : A →C Example 5.5 Let A = {a, b, c, d, e}, B = {α, β, γ, δ} and C = {1, 2, 3, 4, 5, 6}. Let f : A → B be defined by {(a, β), (b, α), (c, δ), (d, δ), (e, δ)}. and g : B → C be the function defined by The composite function, g ◦ f : A →C, is given by Composite Functions

4. 03 May 2009Instructor: Tasneem Darwish4 the function g ◦ f is a subset of the Cartesian product A×C If Let f : A → B and g : D → C, then g ◦ f is defined if and only if the image set of f is a subset of the domain of g Composite Functions

5. 03 May 2009Instructor: Tasneem Darwish5 Examples 5.6 (2) Let f and g be the functions R → R defined by f (x) = x + 2 and g(x) = 1/(x 2 + 1). Then g ◦ f : R →R is defined by And f ◦ g : R →R is defined by in general, f ◦ g ≠ g ◦ f Composite Functions

6. 03 May 2009Instructor: Tasneem Darwish6 Examples 5.6 (3) Three functions, f, g and h, are defined by Determine which of the following composite functions are defined (i)g ◦ f (ii) f ◦ g (iii) h ◦ f Since im( f ) ⊆ R, h ◦ f is defined Composite Functions

7. 03 May 2009Instructor: Tasneem Darwish7 Composite Functions

8. 03 May 2009Instructor: Tasneem Darwish8 In an injective function no two elements of the domain have the same image in the codomain In a surjective function every element of the codomain is the image of at least one element of the domain Injections and Surjections

9. 03 May 2009Instructor: Tasneem Darwish9 Examples 5.7 Let f : {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5, 6}, be defined by and let g : {2, 4, 6, 8, 10, 12} → {2, 3, 5, 7, 11}, be defined by  The function f is injective because each element of the domain has a different image  but f is not surjective because some elements of codomain are not an image for any of the domain elements  The function g is surjective because each element of the codomain is an image for at least one element in the domain  but g is not injective because some elements of the codomain are images for more than one element in the domain Injections and Surjections

10. 03 May 2009Instructor: Tasneem Darwish10 Injections and Surjections

11. 03 May 2009Instructor: Tasneem Darwish11 Example 5.8 The graphs of four functions A → B are given below. Determine whether or not each function is injective and/or surjective. (a) is injective and not surjective (b) is surjective but not injective (c) is both injective and surjective (d) is neither injective nor surjective. Injections and Surjections

12. 03 May 2009Instructor: Tasneem Darwish12