Diese Präsentation wurde erfolgreich gemeldet.

Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
Anzeige
×

1 von 54 Anzeige

# 9-Functions.pptx

sss

Anzeige
Anzeige

## Weitere Verwandte Inhalte

Anzeige

### 9-Functions.pptx

1. 1. Discrete Structures Functions Dr. Muhammad Humayoun Assistant Professor COMSATS Institute of Computer Science, Lahore. mhumayoun@ciitlahore.edu.pk https://sites.google.com/a/ciitlahore.edu.pk/dstruct/ A lot of material is taken from the slides of Dr. Atif and Dr. Mudassir 1
2. 2. Recall the Cartesian Product • All ordered n-tuples (2 tuples in our example) • Let S = { Ali, Babar, Chishti } and G = { A, B, C } • S×G = { (Ali, A), (Ali, B), (Ali, C), (Babar, A), (Babar, B), (Babar, C), (Chishti , A), (Chishti , B), (Chishti , C) } –A relation • The final grades will be a subset of this: –{ (Ali, C), (babar, B), (Chishti, A) } 2
3. 3. Grade Assignment Ali A Babar B Chishti C 3 Ali A Babar B Chishti C
4. 4. Function • This assignment is an example of a function • A function is a set of ordered pairs in which each x-element has only ONE y-element associated with it • The concept of a function is extremely important in mathematics and computer science 4
5. 5. Definition 1 Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f (a) = b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f : A → B. 5
6. 6. Specifying a Function Many different ways: • Sometimes we explicitly state the assignments, as in previous figure • Often we give a formula, such as f (x) = x + 1, to define a function • Other times we use a computer program to specify a function 6
7. 7. 7
8. 8. 8
9. 9. Definition 2 • If f is a function from A to B, we say that A is the domain of f and B is the codomain of f. 9 A B f 4.3 4 Domain Co-domain f(4.3)
10. 10. Definition 2 • If f (a) = b, we say that b is the image of a and a is a preimage of b. 10 R Z f 4.3 4 Domain Co-domain Pre-image of 4 Image of 4.3 f(4.3)
11. 11. Definition 2 • If f is a function from A to B, we say that f maps A to B. 11 R Z f 4.3 4 Domain Co-domain Pre-image of 4 Image of 4.3 f maps R to Z f : A → B f(4.3)
12. 12. Examples 1 2 3 4 5 “a” “bb“ “cccc” “dd” “e” A string length function A B C D F Ali Babar Chishti Dawood Ammara A class grade function Domain Co-domain A pre-image of 1 The image of “a” g(Ali) = A g(Babar) = C g(Chishti) = A … f(x) = length x f(“a”) = 1 f(“bb”) = 2 …
13. 13. Definition 2 • The range of f is the set of all images of elements of A. 13 1 2 3 4 5 a e i o u Some function… Range
14. 14. Not a valid function! 1 2 3 4 5 “a” “bb“ “cccc” “dd” “e”
15. 15. EXAMPLE 1 at Page# 140 • What are the domain, co-domain, and range of the function that assigns grades to students? • Let G be the function that assigns a grade to a student in our discrete mathematics class. • Domain of G is {Adams, Chou, Goodfriend, Rodriguez, Stevens}, • Co-domain is the set {A,B,C,D, F}. • Range of G is the set {A,B,C, F}, 15
16. 16. EXAMPLE 2 at Page# 140 • Let R be the relation with ordered pairs (Abdul, 22), (Brenda, 24), (Carla, 21), (Desire, 22), (Eddie, 24), and (Felicia, 22). Here each pair consists of a graduate student and this student’s age. Specify a function determined by this relation. 16 22 24 21 Abdul Brenda Carla Desire Eddie Felicia If f is a function specified by R, then f (Abdul ) = 22, f (Brenda) = 24, f (Carla) = 21, f (Desire) = 22, f (Eddie) = 24, and f (Felicia) = 22. (Here, f (x) is the age of x, where x is a student.) Domain: set {Abdul, Brenda, Carla, Desire, Eddie, Felicia}. Co-domain: set {21, 22, 24}. Range: set {21, 22, 24}.
17. 17. Example#4 at Page 40 • Let f : Z → Z assign the square of an integer to this integer • What is f (x) =? – f(x) = x2 • What is domain of f ? – Set of all integers • What is codomain of f ? – Set of all integers • What is the range of f ? – {0, 1, 4, 9, . . . }. All integers that are perfect squares 17
18. 18. Function arithmetic • Just as we are able to add (+), subtract (-), multiply (×), and divide (÷) two or more numbers, we are able to + , - , × , and ÷ two or more functions • Let f and g be functions from A to R. Then f + g, f – g, f × g and f/g are also functions from A to R defined for all x ∈ A by: • (f + g)(x) = f(x) + g(x) • (f - g)(x) = f(x) - g(x) • (f g)(x) = f (x)g(x) (f g)(x) Ξ (f × g)(x) • (f/g)(x) = f(x)/g(x) given that g(x)≠0 18
19. 19. Example 6 at Page# 141 • Let f1 and g be functions from R to R such that: • f(x) = x2 //square function • g (x) = x − x2 //some other function • What are the functions f + g and f g? • f + g = (f + g)(x) = f (x) + g(x) = x2 + (x − x2) = x • (f g) = (f g)(x) = f(x)g(x) = x2(x − x2) = x3 − x4 • What is f(x)+g(x) and f+g(x) if x=2? • f(2)=4, g(2)=-2; f(2)+g(2) = 4-2=2 • f+g(2) = 2 19
20. 20. Another Example • Let f and g be functions from R to R such that: • f (x) = 3x+2 • g (x) = -2x + 1 • What is the function f g? • f g= (f g)(x) = f (x)g(x) = (3x+2)(-2x+1) = -6x2- x +2 Let x = -1, what is f(-1).g(-1) and (f g)(-1)? 20 f (-1) = 3(-1) + 2 = -1 g(-1) = -2(-1) + 1= 3 f(-1) g(-1) = -1×3 = -3 (f g) (-1) = -6(-1)2 – (-1) + 2 = -6+1+2 = -3
21. 21. Types of Function • One to One Functions Function, f: X→Y is one-one, if images of distinct elements of X are distinct under f. • One to Many Functions  Function, f: X→Y is one-many, if images of distinct elements of X are not distinct under f. 21 1 2 3 4 5 a e i o A one-to-one function 1 2 3 4 5 a e i o A one-to-many function ( not one-to-one) X Y X Y
22. 22. One-to-one functions • A function is one-to-one if each element in the co-domain has a unique pre-image • Formal definition: A function f is one-to-one if f(x) = f(y) implies x = y. 22 1 2 3 4 5 a e i o A one-to-one function 1 2 3 4 5 a e i o A function that is not one-to-one
23. 23. More on one-to-one • Injective is synonymous with one-to-one – “A function is injective” • A function is an injection if it is one-to-one • Note that there can be un-used elements in a co-domain 23 1 2 3 4 5 a e i o A one-to-one function
24. 24. Example# 9 at Page# 142 • Determine that the function f(x) = x2 of type from (the set of integers to the set of integers is) Z × Z is one-to-one. • 0 -> 0 • 1 -> 1 -1 -> 1 • 2 -> 4 -2 -> 4 • 3 -> 9 -3 -> 9 • 10 -> 100 -10 -> 100 • The function f (x) = x2 is not one-to-one 24
25. 25. Example# 10 at Page# 142 • Determine whether the function f (x) = x + 1 from the set of real numbers to itself is one-to one. • 0 -> 1 • 1 -> 2 • 2 -> 3 • 3 -> 4 • 10 -> 11 • The function f (x) = x + 1 is a one-to-one function. 25
26. 26. • Next Class 26
27. 27. Onto functions • A function is onto if each element in the co- domain is an image of some pre-image • Formal definition: A function f is onto if for all y  C, there exists x  D such that f(x) = y. 27 1 2 3 4 5 a e i o A function that is not onto 1 2 3 4 a e i o u An onto function
28. 28. More on onto • Surjective is synonymous with onto – “A function is surjective” • A function is a surjection if it is onto • Note that there can be multiple used elements in the co-domain 28 1 2 3 4 a e i o u An onto function
29. 29. Example # 12 at Page# 143 • Let f be the function from {a, b, c, d} to {1, 2, 3} defined by f (a) = 3, f (b) = 2, f (c) = 1, and f (d) = 3. Is f an onto function? • f (a) = 3 • f (b) = 2 • f (c) = 1 • f (d) = 3 • Yes, f is an onto function 29
30. 30. Example # 13 at Page# 143 • Determine that the function f(x) = x2 of type Z × Z is onto? • No 30 0 1 2 3 4 5 6 7 8 9 ⁞ 0 1 2 3 ⁞
31. 31. Example # 14 at Page# 143 • Is the function f (x) = x + 1 from the set of integers to the set of integers onto? • 0 -> 1 • 1 -> 2 • 2 -> 3 • 3 -> 4 • 10 -> 11 • The function f (x) = x + 1 is a onto function. 31
32. 32. Onto vs. one-to-one • Are the following functions onto, one-to-one, both, or neither? 32 1 2 3 4 a b c 1 2 3 a b c d 1 2 3 4 a b c d 1 2 3 4 a b c d 1 2 3 4 a b c 1-to-1, not onto Onto, not 1-to-1 Both 1-to-1 and onto Not a valid function Neither 1-to-1 nor onto A) B) C) D) E)
33. 33. Bijections • Consider a function that is both one-to-one and onto: • Such a function is a one-to- one correspondence, or a bijection 33 1 2 3 4 a b c d
34. 34. Example # 16 at Page# 144 • Let f be the function from {a, b, c, d} to {1, 2, 3, 4} with f (a) = 4, f (b) = 2, f (c) = 1, and f (d) = 3. Is f a bijection? • f (a) = 3 • f (b) = 2 • f (c) = 1 • f (d) = 3 • Yes, f is an onto function and one to one function. Hence, Bijection. 34
35. 35. Identity functions • A function such that the image and the pre- image are ALWAYS equal • f(x) = 1*x • f(x) = x + 0 • The domain and the co-domain must be the same set 35
36. 36. Inverse of a Function • For bijections f:AB, there exists an inverse of f, written f 1:BA, which is the unique function such that: • If the inverse function of f exists, f is called invertible. • The function is not invertible if it is not bijection. 36 I f f    1
37. 37. Inverse functions 37 f 4.3 8.6 Let f(x) = 2*x f-1 f(4.3) f-1(8.6) Then f-1(x) = x/2 If f(a) = b, then f-1(b) = a
38. 38. More on inverse functions • Can we define the inverse of the following functions? • An inverse function can ONLY be defined on a bijection 38 1 2 3 4 a b c 1 2 3 a b c d • What is f-1(2)? • Not onto! • What is f-1(2)? • Not 1-to-1!
39. 39. Example 18 at Page #146 • Let f be the function from {a, b, c} to {1, 2, 3} such that f (a) = 2, f (b) = 1, and f (c) = 3. Is f invertible, and if it is, what is its inverse? 39
40. 40. Example 19 at Page #146 • Let f : Z → Z be such that f (x) = x + 1. Is f invertible, and if it is, what is its inverse? • 0 -> 1 • 1 -> 2 • 2 -> 3 • 3 -> 4 • 10 -> 11 • The function f (x) = x + 1 is a one-to-one and onto function, therefore, f is invertible. • Suppose That y=x+1 • x= y-1 • f-1 (y)=y-1 40
41. 41. Example 20 at Page #146 • Let f be the function from R to R with f (x) = X2, Is f invertible? • 0 -> 0 • 1 -> 1 -1 -> 1 • 2 -> 4 -2 -> 4 • 3 -> 9 -3 -> 9 • 10 -> 100 -10 -> 100 • The function f (x) = x2 is not one to one • Therefore, Not Invertible. 41
42. 42. Example 21 at Page #146 • Show that if we restrict the function f (x) = X2 in Example 20 to a function from the set of all nonnegative real numbers to the set of all nonnegative real numbers, then f is invertible. 42
43. 43. Compositions of functions 43 g f f ○ g g(1) f(5) (f ○ g)(1) g(1)=5 f(g(1))=13 1 R R R Let f(x) = 2x+3 Let g(x) = 3x+2 f(g(x)) = 2(3x+2)+3 = 6x+7
44. 44. Compositions of functions Does f(g(x)) = g(f(x))? Let f(x) = 2x+3 Let g(x) = 3x+2 f(g(x)) = 2(3x+2)+3 = 6x+7 g(f(x)) = 3(2x+3)+2 = 6x+11 Function composition is not commutative! 44 Not equal!
45. 45. Proving a function is 1-1 https://www.youtube.com/watch?v=bjATxNZp4GI • A function is said to be 1-1 if whenever F(x)=f(y) then x=y, i.e., for same input, output is also same. 1/30/2023
46. 46. Proving a function is onto https://www.youtube.com/watch?v=Uzlj6N5OYcM 1/30/2023
47. 47. Example 22 at Page# 147 • Let g be the function from the set {a, b, c} to itself such that g(a) = b, g(b) = c, and g(c) = a. Let f be the function from the set {a, b, c} to the set {1, 2, 3} such that f (a) = 3, f (b) = 2, and f (c) = 1. What is the composition of f and g, and what is the composition of g and f ? • Solution:  The composition f ◦ g is defined by (f ◦ g)(a) = f (g(a)) = f (b) = 2,  (f ◦ g) (b) = f (g(b)) = f (c) = 1,  and (f ◦ g)(c) = f (g(c)) = f (a) = 3. • Note that g ◦ f is not defined, because the range of f is not a subset of the domain of g. 47
48. 48. Example 23 at Page# 147 • Let f and g be the functions from the set of integers to the set of integers defined by f (x) = 2x + 3 and g(x) = 3x + 2. What is the composition of f and g? What is the composition of g and f ? • Solution: • Both the compositions f ◦ g and g ◦ f are defined. Moreover, • (f ◦ g)(x) = f (g(x)) = f (3x + 2) = 2(3x + 2) + 3 = 6x + 7 and • (g ◦ f )(x) = g(f (x)) = g(2x + 3) = 3(2x + 3) + 2 = 6x + 11. 48
49. 49. Proving Function problems
50. 50. Defining Functions • Square Function: F: Z→Z F(x)= x*x • Sum Function: Sum: R→R→R Sum (x,y)= x+y • Abs Function: Abs: Z→Z  │x│= 50
51. 51. Defining Functions … • abs function: – 𝑎𝑏𝑠 ∶ 𝑍 ⇒ 𝑍 – 𝑎𝑏𝑠(𝑥) = −𝑥, if 𝑥 < 0 𝑥, otherwise 𝑥 ≥ 0 . • Is the following a valid absolute function? – 𝑎𝑏𝑠 ∶ 𝑍 ⇒ 𝑍 – 𝑎𝑏𝑠(𝑥) = −𝑥, if 𝑥 < 0 𝑥, otherwise 𝑥 > 0 . 51
52. 52. Defining Recursive Function • Factorial of n = n× (n-1) ×(n-2)×…. ×1 • Factorial of 0 = 1 • Factorial of 1 = 1 • 𝑓𝑎𝑐𝑡: 𝑍 ⇒ 𝑍 • 𝑓𝑎𝑐𝑡(𝑥) = 1, if 𝑥 ≤ 0 𝑥 × 𝑓𝑎𝑐𝑡 𝑥 − 1 , 𝑖𝑓 𝑥 > 0. 52
53. 53. Another Example • Suppose that 𝑓 is defined recursively by • 𝑓(0) = 100, • 𝑓(𝑥 + 1) = 𝑓(𝑥) + 3 • Find 𝑓 2 . • To find f(2), we also need to find f(1): • f(1) = f(0) + 3 = 100 + 3 = 103 • f(2) = f(1) + 3 = 103 + 3 = 106 • What is f(5)? • F(5) = f(4)+3 = [f(3)+3] +3 = [[f(2)+3]+3]+3 • = [[[f(1)+3]+3]+3]+3 = [[[[f(0)+3]+3]+3]+3]+3 53
54. 54. 54