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CS 104: Discrete Mathematics

CS 104: Discrete Mathematics. Chapter 3: Fundamental Structures. Functions (In Book: Chapter 2-sec 2.3). Introduction. In many instances we assign to each element of a set a particular element of a second set .

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CS 104: Discrete Mathematics

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  1. CS 104: Discrete Mathematics Chapter 3: Fundamental Structures

  2. Functions (In Book: Chapter 2-sec 2.3) T. Mai Al-Ammar

  3. Introduction • In many instances we assign to each element of a set a particular element of a second set. • E.g. if we have a set of students S={Amal, Rawan, Maha, Sara, Safia} and a set of grads G = { A, B, C, D, F} • we can assigna grade for each student, this assignment is an example • of a function. A B C D F Amal Rawan Maha Sara Safia S G T. Mai Al-Ammar

  4. Functions • Definition: • Let A and B be nonempty sets. A function ffrom A to B( A  B ) is an assignmentof 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. • Function are sometimes also called mappingor transformations. • Functions are specified in different ways : • Explicitly state the assignment ( as example in the previous slide) • Give a formula, e.g. • Use a computer program to specify a function • Define a function in terms of a relation from A to B T. Mai Al-Ammar

  5. Functions ( Cont.) • Definition: • If the function f : A  B, then A is the domain and B is the co-domain of f. • If f(a) = b, we say that b is the image of a and a is a pre-image of b. • The range or image of f is the set of all images of elements of A. • If f is a function from A to B, we say that fmaps A to B. a A b=f(a) B f(a) f T. Mai Al-Ammar

  6. Functions ( Cont.) • To define a function, we specify its domain, codomain, and the mapping of elements of the domain to elements in the codomain. • Two functions are equalwhen they have the same domain, have the same codomain, and map each element in the domain to the same element in the codomain . • If we change a domain or codomain or a mapping of a function, we obtain a different function. T. Mai Al-Ammar

  7. Functions ( Cont.) Example: What are the domain, co-domain and range of the function that assigns grades to students? Let G be the function that assigns grades to a student in our class. The domain is the set {Amal, Rawan, Maha, Sara, Safia} The co-domain is the set {A, B, C, D, F} The range of G is the set {A, B, C, F} A B C D F Amal Rawan Maha Sara Safia S G T. Mai Al-Ammar

  8. Functions ( Cont.) Example : Let R be the relation with ordered pairs (Amal,22), (Bdoor, 24), (Sara, 21), (Dalal, 22), (Eman, 24) and (Fadwa, 22). Here each pair consists of a graduate student and this student’s age. Specify a function determined by this relation. If f is a function specified by R, then : f(Amal)=22, f(Bdoor)= 24, f(Sara)=21, f(Dala)=22, f(Eman)= 24, f(Fadwa)=22 Here f(X) is the age of x, where x is a student. Domain {Amal, Bdoor, Sara, Dalal, Eman, Fadwa} Co-domain { a | a>10 and a<90} Range {21,22,24} T. Mai Al-Ammar

  9. Functions ( Cont.) Example: Let f be the function that assigns the last two bits of a bit string of length 2 or greater to that string. For example, f(11010)=10. The domain of f is the set of all bit strings of length 2 or greater both the co-domain and the range are the set {00, 01, 10, 11} Example: Let f: ZZ assign the square of an integer to this integer. Domain and Co-domain of f is the set of all integers Range of f is the set of all integers that are perfect squares {0, 1, 4, 9, …} T. Mai Al-Ammar

  10. Functions ( Cont.) The domain and codomain of functions are often specified in computer programs e.g. the domain of the function is real numbers ( float ) , and the codomain of the function is integers. intMyFunction ( float x ) T. Mai Al-Ammar

  11. Functions ( Cont.) Definition: Let f1and f2be two functions from A to R. Then f1+ f2and f1 f2are also functions from A to R defined by (f1+f2) (x) = f1(x) + f2(x) (f1 f2)(x) = f1 (x) f2(x) Example : Let f1and f2be two functions from R to R such that f1 (x) = x2and f2 (x) = x – x2. What are the functions f1+f2 , f1 f2? (f1+f2)(x) = f1 (x) + f2 (x) = x2 + (x – x2) = x (f1f2)(x) = f1 (x) f2(x) = x2(x – x2) = x3– x4 T. Mai Al-Ammar

  12. Functions ( Cont.) A function f from the set A to the set B is said to be one-to-oneor an injunction, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. (No element of B is the image of more than one element in A) a b ( f (a) = f (b)  a=b ) Taking the contrapositive of the implication in the definition: a b (a ≠ b  f (a) ≠ f (b)) Every b  B has at most one preimage. T. Mai Al-Ammar

  13. Functions ( Cont.) Example : Determine whether the function f: {a, b, c, d}  {1, 2, 3, 4, 5} with f(a) = 4, f(b) = 5, f(c) = 1, and f(d) = 3 is injective. The function f is one-to-one since f takes on different values at the four elements of its domain. Example : Determine whether the function f : Z  Z , f (x) = x2is injective. The function f is not one-to-one since, for instance, f(1) = f(-1) = 1, but 1 ≠ -1. Example : Determine whether the function f (x) = x+1 from the set of real numbers to itself is one-to-one. T. Mai Al-Ammar

  14. Functions ( Cont.) • A function f from the set A to the set B is said to be ontoor surjection , if and only if for every element b  B, there is an element a  A with f (a) = b. (All elements in B are used.) • Every b  Bhas at least one preimage. T. Mai Al-Ammar

  15. Functions ( Cont.) Example : Determine whether the function f: {a, b, c, d}  {1, 2, 3} with f(a) = 3, f(b) = 2, f(c)= 1, and f(d) = 3 is surjective. The function f is onto, since three elements of the co-domain are images of elements in the domain. Example: Determine whether the function f: Z  Z, f(x) = x2is surjective. The function f is not onto since, e.g. there is no integer x with x2 = -1. Example : Determine whether the function f (x) = x+1 from the set of integers to itself is onto. T. Mai Al-Ammar

  16. Functions ( Cont.) Functions can be both one-to-one and onto. Such functions are called bijective. Bijections are functions that are both injective and surjective. Every b  B has exactly one preimage. T. Mai Al-Ammar

  17. Example: Determine whether the function f: {a, b, c, d}  {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d) = 3 is bijective. The function f is one-to-one since no two values in the domain are assigned the same function value. It is also onto because all four elements of the co domain are images of elements in the domain. Hence, f is a bijection. T. Mai Al-Ammar

  18. Some examples T. Mai Al-Ammar

  19. Relations(In Book: Chapter 9- sec 9.1 , sec 9.2) T. Mai Al-Ammar

  20. Relations The most direct way to express a relationship between elements of two sets is to use ordered pairs made up of two related elements. A set of ordered pairs are called binary relations. Definition: Let A and B be sets. A binary relation from A to B is a subset of A x B. A binary relation from A to B is a set R of ordered pairs where the first element of each ordered pair comes from A and the second element comes from B. a R b (a,b)  R a R b (a,b) ∉R T. Mai Al-Ammar

  21. Relations ( Cont.) Example: Let A = Set of students; B = Set of courses, let R be the relation that consists of pairs : R = {(a,b) | student a is enrolled in course b} • Note that when a student is not enrolled in any course, there will be no pairs in Rthat have this student as the first element. Example : Let A = Set of cities; B = Set of countries. Define the relation R by specifying that (a, b) belongs to R if city a is the capital of b. For instance, (Riyadh, Saudi Arabia), (Delhi, India), (Washington, USA) are in R. T. Mai Al-Ammar

  22. Relations ( Cont.) Example: Let A={0, 1, 2} and B={a, b}. {(0, a), (0, b), (1, a), (2, b)} is a relation from A to B. This means, 0Ra, but 1Rb. • Relations can be represented in two ways - as shown in the figure: • Graphically using arrows to represent ordered pairs. • Using a table 0. .a .b 2. T. Mai Al-Ammar

  23. Relations ( Cont.) A function f from a set A to B assigns exactly one element of B to each element of A. Every element of A is the first element of exactly one ordered pair in the relation. A relation can be used to express a one to many relationships between the elements of the sets A and B, i.e. an element of A may be related to more than one element of B. A function represents a relation where exactly one element of B is related to each element of A, i.e. every element has only one image. T. Mai Al-Ammar

  24. Relations ( Cont.) Definition: A relation on the set A is a relation from A to A. That is, a relation on a set A is a subset of A x A. Example: Let A = {1, 2, 3, 4}. Which ordered pairs are in the relation R={(a, b) | a divides b}? (a, b) є R if and only if a and b are positive integers not exceeding 4 such that a divides b, we see that R={(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)} The pairs in R are displayed graphically and in tabular form: T. Mai Al-Ammar

  25. Relations ( Cont.) The pairs in R are displayed graphically and in tabular form: 1 .1 2. .2 3. .3 4. .4 T. Mai Al-Ammar

  26. Relations ( Cont.) Example : Consider the relations on the set of integers: R1= {(a, b) | a ≤ b}, R2={(a, b) | a > b}, R3={(a, b) | a = b or a = -b}, R4={(a, b) | a = b}, R5={(a, b) | a = b+1}, R6={(a, b) | a+b ≤ 3}, Which of these relations contain each of the pairs (1, 1), (1, 2), (2, 1), (1, -1) and (2, 2)? (1, 1) is in R1, R3, R4 and R6; (1, 2) is in R1and R6; (2, 1) is in R2, R5 and R6; (1, -1) is in R2, R3 and R6; (2, 2) is in R1, R3 and R4. T. Mai Al-Ammar

  27. Relations ( Properties of Relations ) 1. Reflexivity: A relation R on a set A is called reflexive if (a, a)  R for all a  A. 2. Symmetry: A relation R on a set A is called symmetric if (b, a)  R whenever (a, b)  R, for all a, b  A. 3. Antisymmetry: A relation R on a set A is called antisymmetric if for all a, b  A, if (a, b)  R and (b, a)  R, then a = b. The contrapositive is : T. Mai Al-Ammar

  28. Relations ( Properties of Relations ) 4. Transitivity: A relation R on a set A is called transitive if (a, b)  R and (b, c)  R imply (a, c)  R, for all a, b, c  A. Remark: The terms symmetric and antisymmetricare not opposites. A relation can have both of these properties or may lack both of them. A relation can not be both symmetric and antisymmetric if it contains some pair of the form (a,b) where a b In antisymmetric, the only way to have a related to b and b related to a is for a and b to be the same element. T. Mai Al-Ammar

  29. Relations ( Properties of Relations ) Example : Which of the following relations are reflexive and symmetric? Consider the relations on the set of integers: R1= {(a, b) | a ≤ b}, R2={(a, b) | a > b}, R3={(a, b) | a = b or a = -b}, R4={(a, b) | a = b}, R5={(a, b) | a = b+1}, R6={(a, b) | a+b ≤ 3}, The reflexive relations are R1(because a ≤ a, for all integer a), R3 and R4. For each of the other relations ,it is easy to find a pair of the form (a, a) that is not in the relation. T. Mai Al-Ammar

  30. Relations ( Properties of Relations ) The symmetric relations are R3, R4 and R6. R3 is symmetric, for if a=b or a=-b, then b=a or b=-a. R4 is symmetric, since a=b implies b=a. R6 is symmetric, since a+b ≤ 3 implies b+a ≤ 3. None of the other relations is symmetric. T. Mai Al-Ammar

  31. Relations ( Properties of Relations ) Example : Which of the following relations are antisymmetric? Relations of integers R1= {(a, b) | a ≤ b}, R2={(a, b) | a > b}, R3={(a, b) | a = b or a = -b}, R4={(a, b) | a = b}, R5={(a, b) | a = b+1}, R6={(a, b) | a+b ≤ 3}, R1 is antisymmetric, since the inequalities a ≤ b and b ≤ a imply that a = b. R2is antisymmetric, since it is impossible for a>b and b>a. R4 is antisymmetric because two elements are related with respect to R4 if and only if they are equal. R5 is also antisymmetric, since it is impossible that a = b+1 and b = a+1. T. Mai Al-Ammar

  32. Relations ( Properties of Relations ) Example: Which of the following relations are transitive? Relations of integers R1= {(a, b) | a ≤ b}, R2={(a, b) | a > b}, R3={(a, b) | a = b or a = -b}, R4={(a, b) | a = b}, R5={(a, b) | a = b+1}, R6={(a, b) | a+b ≤ 3}, The transitive relations from Example 5 are R1, R2, R3 and R4. R1 is transitive, since a ≤ b and b ≤ c imply a ≤ c. R2 is transitive, since a > b and b > c imply a > c. R3 is transitive, since a = ±b and b = ±c imply a = ±c. R4 is transitive, since a = b and b = c imply a = c. T. Mai Al-Ammar

  33. Relations ( Properties of Relations ) Example : Consider following relations on {1, 2, 3}: R1= {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3)}, R2={(1, 1), (1, 2), (2, 2), (3, 2), (3, 3)}, R3={(2, 1), (2, 3), (3, 1)} R4={(2, 3)}, Which of the relations are reflexive, symmetric, antisymmetric and transitive? T. Mai Al-Ammar

  34. Relations ( Properties of Relations ) Definition: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. 1. Reflexive ( aA, aRa) 2. Symmetric (aRb => bRa) 3. Transitive (aRb and bRc => aRc) T. Mai Al-Ammar

  35. Relations ( Properties of Relations ) Example : Let R be the relation on the set of real numbers such that aRb if and only if a-bis an integer. Is R an equivalence relation? As a-a = 0 is an integer for all real numbers a. So, aRa for all real numbers a. Hence R is reflexive. Let aRb, then a-b is an integer, so b-a also an integer. Hence bRa, i.e., R is symmetric. If aRb and bRc, then a-b and b-c are integers. So, a-c = (a-b) + (b-c) is also an integer. Hence, aRc. Thus R is transitive. Consequently, R is an equivalence relation T. Mai Al-Ammar

  36. End of Chapter 3 T. Mai Al-Ammar

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