1. Example Prove that: “IF 3n + 2 is odd, then n is odd” Proof by Contradiction: -p = 3n + 2 is odd, q = n is odd. -Assume that ~(p  q) is true OR -(p  ~q) is true. It means that both p AND ~q must be true -To show contradiction, show that q is also true, OR Show that ~p is also true. -~q = n is not odd = n is even -n = 2k -3n + 2 = 6k + 2 = 2 (3k + 1) = 2t is even -~p is true -Thus, p and ~p are both true  contradiction

2. Example

3. Functions Definition: Let X and Y be nonempty sets. A function f from X to Y: for each x  X, there is exactly one y  Y We write f(x) = y is the unique element of Y assigned by the function f to the element x of X. Notation: f : X  Y f is a function from X to Y Other names: mappings, tranformations

4. Example 1 2 3 4 a b c d 1 2 3 4 a b c d 1 2 3 4 a b c d XY XY XY fungsiBukan fungsi

5. Some Terminology Definition: f is a function from A to B. A is the domain of f. B is the codomain of f. f(a) = b. b is the image of a. a is the preimage of b. The range of f is the set of all images of elements of A. f maps A to B. From previous example: f is the function “lahir di” f(x) = lahir di Domain: {xx, yy, zz} Codomain: (Jakarta, Solo, Bogor} The range of f: {Jakarta, Solo}

6. Example R is the relation consist of pairs {Abdul, 22}, (Brenda, 24}, {Carla, 21}. What are the domain, codomain and function? Domain: {Abdul, Brenda, Carla} Codomain: {21, 22, 24} Function f, ages of students Range: {21, 22, 24}

7. Equality of Functions Definition: Two functions are equal when they have the same domain, have the same codomain, and map elements of their common domain to the same elements in their common codomain.

8. Sum and Product of Functions Definition: Let f 1 and f 2 be functions from A to R. Then f 1 +f 2 and f 1 f 2 are also functions from A to R defined by. Notation: (f 1 + f 2 )(x) = f 1 (x) + f 2 (x) (f 1 f 2 )(x) = f 1 (x) f 2 (x)

9. Example f 1 (x) = x 2 f 2 (x) = x – x 2 (f 1 + f 2 )(x) = f 1 (x) + f 2 (x) = x 2 + (x – x 2 ) = x (f 1 f 2 )(x) = f 1 (x) f 2 (x) = x 2 (x – x 2 ) = x 3 – x 4

10. Image of a Subset Definition: Let f be a function from the set A to the set B and let S be a subset of A. The image of S under the function f is the subset of B that consists of the images of the elements of S. we denote the image of S by f(S). Notation: f(S) = {t |  s  S (t = f(s))} {f(s) | s  S}

11. Example A = {a, b, c, d} B = {1, 2, 3, 4} f(a) = 2, f(b) = 1, f(c) = 4, f(d) = 1. S = (b, c, d) f(S) = {1, 4}

12. One-to-One Function Definition: A function f is said to be one-to-one, or injective, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. A function is said to be an injection if it is one- to-one. Notation:  a  b (f(a) = f(b)  a = b)  a  b (a ≠ b  f(a) ≠ f(b))

13. Example A = {a, b, c, d} B = {1, 2, 3, 4, 5} f(a) = 4, f(b) = 5, f(c) = 1, f(d) = 3. Is this one-to-one? Yes, because f takes on different values at the four elements of its domain. f(x) = x 2, x is integer Is this one-to-one? No, because f(1) = f(-1)

14. Example

15. Onto Function Definition: A function f from A to B is called onto, or surjective, if and only if for every elements b  B there is an element a  A with f(a) = b. A function f is called a surjection if it is onto. Notation:  y  x ( f(x) = y)

16. Example A = {a, b, c, d} B = {1, 2, 3} f(a) = 3, f(b) = 2, f(c) = 1, f(d) = 3. Is this onto? Yes, because all elements of B are images of elements of A. f(x) = x + 1, x is integer Is this onto? Yes, because for every integer y there is an x such that f(x) = y.

17. One-to-One Correspondence Definition: The function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto. A = {a, b, c, d} B = {1, 2, 3, 4} f(a) = 4, f(b) = 2, f(c) = 1, f(d) = 3. Is this a bijection? Yes, because it is both one-to-o ne and onto.

18. One-to-One Correspondence One-to-one, not onto Onto, not one-to-one One-to-one, onto Neither one-to-one, nor onto Not a function

19. Inverse Function Definition: Let f be a bijection from the set A to the set B. The inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a) = b. The inverse function of f is denoted by f -1. Hence, f -1 (b) = a when f(a) = b.

20. Example A = {a, b, c} B = {1, 2, 3} f(a) = 2, f(b) = 3, f(c) = 1 Is f invertible? Yes, f -1 (1) = c, f -1 (2) = a, f -1 (3) = b f(x) = x + 1, x is integer Is f invertible? Yes. y = x + 1 x = y – 1 f -1 (x) = y - 1

21. Example f(x) = x 2 Is f invertible? No. Because f(-2) = f(2) = 4

22. Composition of Functions Definition: Let g be a function from the set A to the set B and let f be a function from the set B to set C. The composition of the functions f and g, denoted by f ○ g, is defined by (f ○ g)(a) = f(g(a))

23. Composition of Functions

24. f ○ g = g ○ f?

25. Composition of Functions f(x) = 2x + 3 g(x) = 3x + 2 What are f ○ g and g ○ f? (f ○ g)(x) = f(g(x)) = f(3x + 2) = 6x + 7 (g ○ f)(x) = g(f(x)) = g(2x + 3) = 6x + 11

26. The Graph of Functions Definition: Let f be a function from the set A to the set B. The graph of the function f is the set of ordered pairs {(a, b) | a  A and f(a) = b} Example: F(x) = x 2 X are integer The graph of f is the set of (x, f(x)) = (x, x 2 )

27. Floor and Ceiling Definition: The floor function assigns to the real number x the largest integer that is less than or equal to x. The ceiling function assigns to the real number x the smallest integer that is greater than or equal to x. Notation: floor :  x  ceiling :  x 

28. Example  0.5  = ? 0  0.5  = ? 1  8.9  = ? 8  8.9  = ? 9

29. Floor and Ceiling True or False?  x + y  =  x  +  y   x + y  =  x  +  y 

30. Sequences Sequence is an ordered list of elements. A sequence is a special type of function in which the domain consists of a set of consecutive integers. Definition: A sequence is a function from a subset of the set of integers (usually either the set {0, 1, 2, …} or the set {1, 2, 3, …}) to a set S. a n denote the image of the integer n and called as the term of the sequence. Finite sequence: {1, 2, 3, 4}, {1, 3, 5, 7} Infinite sequence: {1, 2, 3, …}, {1, 3, 5, 7, …}

31. Sequences Sequence {a n }, where a n = 1/n The terms of this sequence: a 1, a 2, a 3, …, Starts with 1, 1/2, 1/3, 1/4, …,

32. Geometric Progression Definition: A geometric Progression is a sequence of the form a, ar, ar 2, …, ar n, … Where the initial term a and the common ratio r are real numbers. Sequence {c n } with c n = 2. 5 n Initial term: 2 Common ratio: 5 List of terms c 0, c 1, c 2, … = 2, 10, 50, …

33. Geometric Progression Sequence {d n } with d n = 6. (1/3) n Initial term: 6 Common ratio: 1/3 List of terms d 0, d 1, d 2, … = 6, 2, 2/3, … Sequence {b n } with b n = (-1) n Initial term: 1 Common ratio: -1 List of terms b 0, b 1, b 2, … = 1, -1, 1, …

34. Arithmetic Progression Definition: A Arithmetic Progression is a sequence of the form a, a + d, a + 2d, …, a + n.d, Where the initial term a and the common difference d are real numbers. Sequence {s n } with s n = -1 + 4n Initial term: -1 Common difference: 4 List of terms s 0, s 1, s 2, s 3, … = -1, 3, 7, 11, …

35. Arithmetic Progression Sequence {t n } with t n = 7 – 3n Initial term: 7 Common difference: -3 List of terms t 0, t 1, t 3, … = 7, 4, 1, -2, …

36. Special Integer Sequences How to find a general rule or formula for constructing the terms of a sequence? 1, 1/2, 1/4, 1/8, 1/16 a n = 1/2 n  geometric progression 1, 3, 5, 7, 9 a n = 2n + 1  arithmetic progression 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …. a 11 = …, a 12 = … 5, 11, 17, 23, 29, … a n = 6n – 1  Compare with the well-known integer sequence

37. Well-Known Integer Sequences Check On-Line Encyclopedia of Integer Sequences

38. Special Integer Sequences 1, 7, 25, 79, 241 a n = 3 n – 2 0, 4, 18, 48, 100, … a n = n 3 – n 2

39. Summations Sequence: a m, a m+1, … a n Summation: a m + a m+1 + … a n

40. Summations The sum of the first 100 terms of the sequence {an} where a n = 1/n

41. String A string is a finite sequence of characters. Example: “Saya kuliah Matematika Diskrit” A string over X, where X is a finite set, is a finite sequence of elements from X. Example: X = {a, b, c}  1 = b  2 = a  3 = a  4 = c The string over X: baac baac ≠ abac baac = ba 2 c : string with no elements or null string

42. String The length of a string  is the number of elements in .  = aabab  = a 3 b 4 a 32 |  | = 5|  | = 39 Concatenation of  and  :  = aababa 3 b 4 a 32 Substring : some or all consecutive elements of .  = aba is a substring of .