RELATIONS, FUNCTIONS AND COUNTING PROBLEMS 1 Relations Definition 1 A relation is a set of ordered pairs Example 1 The follows are relations a) “Less than” = {(1,2), (3,4), (5,6), } b) “Greater than” = {(2 1,2), (5 3,4 5), (5 1,2 0), } c) “equal to” = {(1,1), (2,2), (3,3), } > If R is a relation, pair (x,y) ∈ R can be written as xRy or “x is related to y by the relation R” Exercise 1 Write the following relations on the set {1, 2, 3, 4, 5} as sets of ordered pairs a) The is less than relation b).
RELATIONS, FUNCTIONS AND COUNTING PROBLEMS Relations Definition A relation is a set of ordered pairs Example The follows are relations a) “Less than” = {(1,2), (3,4), (5,6), } b) “Greater than” = {(2.1,2), (5.3,4.5), (5.1,2.0), } c) “equal to” = {(1,1), (2,2), (3,3), } If R is a relation, pair (x, y) ∈ R can be written as xRy or “x is related to y by the relation R” Exercise Write the following relations on the set {1, 2, 3, 4, 5} as sets of ordered pairs a) The is-less-than relation b) The is-divisible-by relation c) The is-equal-to relation d) The has-the-same-parity-as relation Definition Let R be a relation and let A and B be sets We say R is a relation on A Iff R ⊆ A × A, and we say R is a relation from A to B Iff R⊆A×B Example Let A={1,2,3}, B={4,5,6,7} a) R= {(1,1), (2,2), (3,3)} on A b) S= {(1,4), (2,6), (3,5)} A→B c) T= {(5,4), (7,6), (6,5),(7,5)} d) X= {(5,9), (8,6)} on B not A→B / B→A 1.1 Inverse relations Definition Let R be a relation The inverse of R, denoted R−1 , is the relation formed by reversing the order of all the ordered pairs in R R−1 = {(x, y) : (y, x) ∈ R} : (R−1 )−1 = R Example Let S = {(1, 4), (2, 6), (3, 5)} S −1 = {(4, 1), (6, 2), (5, 3)} 1.2 Properties of relations Definition Let R be a relation on A i) If ∀x ∈ A ⇒ xRx, we call R reflexive ii) If ∀x ∈ A ⇒ x Rx, we call R irreflexive iii) If ∀x, y ∈ A, xRy ⇒ yRx, we call R symmetric iv) If ∀x, y ∈ A, (xRy ∧ yRx) ⇒ x = y, we call R antisymmetric v) If ∀x, y, z ∈ A, (xRy ∧ yRz) ⇒ xRz, we call R transitive Example Let’s consider the relation = (equality) on the integers Z a) Is the relation “=” reflexive? ✓ b) Is the relation “=” irreflexive? ✗ c) Is the relation “=” symmetric? ✓ d) Is the relation “=” antisymmetric? ✓ e) Is the relation “=” transitive? ✓ 1.3 Equivalence relations Definition Let R be a relation on a set A We say R is an equivalence relation Iff it is reflexive, symmetric and transitive Definition Let n be a positive integer We say x and y are congruent modulo n, and write x ≡ y(mod n), provided n|(x − y) (or x − y is a multiple of n) Denote “≡” as “is congruent to mod n” Example a) ≡ 23(mod 5), because 3-23=-20 is divisible by b) ≡ 3(mod 5), because 9-3=6 is not a multiple of ≡ is an equivalence relation? Theorem Let n be a positive integer The is-congruent-to-mod-n relation is an equivalence relation on the set of integers Proof : “is-congruent-to-mod-n” is an equivalence relation (i) is reflexive? ∀x ∈ Z, since x − x = 0 ⇒ x ≡ x(mod n) (ii) is symmetric? ∀x, y ∈ Z, and x ≡ y(mod n) ⇒ x − y = kn ⇒ y − x = (−k)n ⇒ y ≡ x(mod n) (iii) is transitive? x ≡ y(mod n) y ≡ z(mon n) x − z = (k + m)n ⇒ x ≡ z(mod n) ∀x, y, z ∈ Z, suppose 1.4 ⇒ x−y = kn and y −z = mn ⇒ Equivalence class Definition Let R be an equivalence on a set A and let a ∈ A The equivalence class of a, denoted [a], is the set of all elements of A related by R to a; that is: [a] = {x ∈ A : xRa} Propositions Let R be an equivalence relation on a set A a) Any a ∈ A, then a ∈ [a] b) Any a, b ∈ A, then aRb iff [a] = [b] c) Any a, x, y ∈ A If x, y ∈ [a] then xRy d) Suppose [a] ∩ [b] = ∅ Then [a] = [b] Example Find the requested equivalence class: a R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)} on {1, 2, 3, 4} Find [1]? b R = “has-the-same-birthday-as” (only days and months) on people Find [you]? Corollary Let R be an equivalence relation on a set A The equivalence classes of R are nonempty, pairwise disjoint of A whose union is A Example How many different classes are there? a Let R be “is-congruent-to-mod-5 ” on Z b Let R be “has-the-same-birthday-as” on set of days and months 1.5 Partitions Definition Let A be a set A partition on A is a set of nonempty, pairwise disjoint sets whose union is A P1 P3 P2 P5 P4 Elements of a partition are called parts Example Find all possible partitions of {1,2,3}? 2 3 2 3 How can you partition people in the World? • live in the same city • live in the same country • have the same birth month • have the same age • Question: What is the relationship between partition and equivalence relation? 1.6 Partition and equivalence relation Propositions Suppose R is a relation on set A i) If R be an equivalence relation, then all the equivalence classes of R form a partition of A ii) If P is a partition on A, we define a relation: P a ≡ b ⇔ ∃P ∈ P|a, b ∈ P, or P ≡ is is-the-same-part-as relation P The relation ≡ is an equivalence relation on A The equivalence classes P of ≡ are exactly the parts of P Example How many possible equivalence relations are there on set {1,2,3,4}? Functions Definition A relation f is called a function iff (a, b) ∈ f , and (a, c) ∈ f imply b = c Pair (a,b) is unique in function f for each a Example 10 Which relation is function? a) f = {(1, 2), (3, 4), (5, 6), (6, 7)} ⇒ is a function b) g = {(1, 2), (1, 3), (3, 2)} ⇒ is not a function 2.1 Function notations Definition 10 Let f be a function, let a be an object The notation f (a) is defined provided there exists an object b such that (a, b) ∈ f In this case, f (a) equals b Otherwise, there is no ordered pair of the form (a, _) ∈ f , the notation f (a) is undefined Example 11 f ={(1, 1), (2, 3), (3, 4)} ⇔{f (1) = 1, f (2) = 3, f (3) = 4, f (4) is undef ined, } Some ways to express functions • as a set of pairs f = {(1, 2), (2, 3), (5, 6)} • as a chart x f(x) 4 5 6 • Express function as a map y1 x1 y2 x2 x3 y3 x4 y4 x5 y5 • Express function as a set-builder notation f = {(x, y) : x, y ∈ Z, y = x2 } 2.2 Domain, image of functions Definition 11 Let f be a function i) The set of all possible first elements of the ordered pairs in f is called the domain of f , denoted dom(f ) dom(f ) = {a : ∃b, (a, b) ∈ f } ii) The set of all possible second elements of the ordered pairs in f is called the image of f , denoted im(f ) im(f ) = {b : ∃a, (a, b) ∈ f } Example 12 Find image and domain of the following functions a) f = {(1, 2), (2, 3), (5, 6)} ⇒ dom(f ) = {1, 2, 5}, im(f ) = {2, 3, 6} b) f = {(x, y) : x, y ∈ Z, y = x2 } ⇒ dom(f ) = Z, 2.3 im(f ) = {0, 1, 4, , n2 , } One-to-one/onto/bijection functions Definition 12 Suppose f is a function from set A to set B i) A function f is called one-to-one provided that, whenever (x, b), (y, b) ∈ f , we must have x = y ii) A function f is called onto provided that, for every b ∈ B there is an a ∈ A so that f (a) = b iii) Let f be a function, f is called a bijection iff it is both one-to-one and onto Example 13 Let A be the set of odd integers, B be the set of even integers The function f : A → B, defined by f (x) = x + • f is one-to-one? • f is onto? Properties Let f : A → B be a function i) The inverse relation f −1 is a function iff f is one-to-one ii) f is onto B iff im(f ) = B iii) The inverse relation f −1 is a function from B to A iff f is one-to-one and onto B Example 14 For each of the following, determine whether the function is one-to-one, onto, or both Prove your assertions a) f : Z → Z, f (x) = 2x b) f : Z → Z, f (x) = x + 10 c) f : N → N, f (x) = x + 10 d) f : N → N defined by f (x) = 2.4 x/2 if x is even (x − 1)/2 if x is odd Composition Definition 13 Let A, B, and C be sets and let f : A → B and g : B → C Composition of g and f , denoted g ◦ f , defined by (g ◦ f )(a) = g[f (a)] Example 15 Let f : Z → Z by f (x) = x2 +1 and g : Z → Z by g(x) = 2x−3 What is g ◦ f (x)? Propositions f −1 ◦ f = f ◦ f −1 If f : A → B is bijection then • (f −1 ◦ f )(a) = a, ∀a ∈ A • (f ◦ f −1 )(b) = b, ∀b ∈ B Collections: Lists and Sets Definition 14 A list is an ordered sequence of objects All the list objects are enclosed in round brackets ( ) Example 16 The follows are lists (i) S=(0,1,2,3) (ii) T=(-2,3,2,3,4) Definition 15 A set is a repetition-free, unordered collection of objects All the set objects are enclosed in curly braces { } Example 17 The follows are sets (i) S={0,1,2,3} (ii) T={3,2,4} (iii) U = {x ∈ N : x ≤ 20 and 6|x} (6|x means is a divisor of x) List and set differences: List Set Object order ✓ neglect Object repetition ✓ neglect Example 18 Notice that a) (0,1,2), (0,2,1) and (0,1,2,1) are pairwise different b) {0,1,2}, {0,2,1} and {0,1,2,1} are the same 3.1 List and Set Size (i) Let A be a list (or set) |A| is denoted as the number of list (or set) elements (ii) If A is a list, |A| is called the length of A (iii) If A is a set, |A| is called the cardinality of A (iv) ( )-empty list, ∅ - empty set Example 19 We have a) S = {0, 1, 2, 3} ⇒ |S| = b) T = {3, 2, 4} ⇒ |T | = 3.2 Equality of lists, sets (i) For two lists L1 , L2 L1 = L2 ⇔ length of L1 = length of L2 L1 [i] = L2 [i], ∀i = 1, 2, (ii) For two sets S1 , S2 S1 = S2 ⇔ 3.3 ∀x ∈ S1 ⇒ x ∈ S2 ∀x ∈ S2 ⇒ x ∈ S1 Counting List Problem Suppose we wish to make a two-element list where the entries in the list may be any in the set S={1, 2, 3, ,n} (i) How many such lists are possible? (ii) In addition, the two numbers on the list must be different How many such lists are possible? (iii) In general, to make k-element lists where the entries in the list are in S and pairwise different How many such lists are possible? Problem Suppose we wish to make a two-element list where the entries in the list may be any in theset S={1, 2, 3, ,n} 1) How many such lists are possible? n2 2) In addition, the two numbers on the list must be different How many such lists are possible? n2 − n = n(n − 1) 3) In general, to make k-element lists where the entries in the list are in S and pairwise different How many such lists are possible? n(n − 1)(n − 2) (n − k + 1) denoted as (n)k 3.4 Subsets Definition 16 Suppose A and B are sets We say that A is a subset of B provided every element of A is also an element of B The notation A ⊂ B means A is a subset of B 10 Notes: (i) A ⊃ B ⇔ B ⊂ A (ii) ∅ ⊂ A, ∀A (iii) A ⊆ B means A is a subset or equal to B Example 20 We have (i) {2, 3} ⊂ {0, 1, 2, 3} (ii) {2n : n ∈ N} ⊂ N 3.5 Counting subsets Problem How many subsets does A = {1, 2, 3} have? Answer: subsets of A ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} Theorem Let A be a finite set The number of subsets of A is 2|A| Notes: (i) We denote 2A as the set of all subsets of A (ii) |2A | = 2|A| 3.6 Exercise Suppose there are 62 students in discrete maths class We have to send a team with at least one student to join a sport event (a) How many ways to choose such group are possible? (b) Suppose all the students are taking part in running or swimming In the list of running attendees, we have 45 students of the class and 35 students in the list of swimming attendees How many students are taking part in both running and swimming? Example 21 Let A be {1,2,3,4,5} a) How many subsets of A are possible? b) How many non-empty subsets of A are possible? 11 c) How many 4-element subsets of A are possible? Theorem Let A be an n-element set The number of k-element subsets of n! A is C(n, k) = (n − k)!.k! 3.7 Counting classes/parts Problem In how many ways (i) can the letters in the word “WORD” be rearranged? (ii) can the letters in the word “DISCRETE ” be rearranged? Theorem Let R be an equivalence relation on a finite set A If all the equivalence classes of R have the same size, m, then the number of equivalence classes is |A|/m Example 22 How many different rearrangements (including nonsensical words that are called anagrams) can be made from SUCCESS? Solution: (i) Let S be the set of all rearrangements of all letters of SUCCESS ⇒ |S| = 7! (ii) Define “is-the-same-rearrangement-as” as a relation R on S (iii) #anagrams = 3.8 7! |S| = an equivalence class size 3!.2! Multisets Definition 17 A multiset is an unordered collection of objects with repetition All the set objects are enclosed in angle braces Example 23 We have a) S= b) T= c) V= 12 Definition 18 (1) The cardinality of a multiset is the sum of multiplicities of its elements S= ⇒ |S| = (2) Two multisets are said the same Iff they contain the same elements with the same multiplicities (i) < 1, 1, 2, >=< 1, 2, 3, > (ii) < 1, 2, 3, >=< 1, 2, 2, > 3.9 Counting multisets Problem How many multisets with cardinality equal to k whose elements belong to an n-element set such as {1,2, ,n}? Example 24 All the multisets with size we can form from the elements in {1,2,3} are: , , , , , Suppose a multiset of size k with elements drawn from {1,2,3, ,n}, ei is the i’s multiplicity We have two correspondences: f < 1, , 1, 2, , 2, , n, , n > ←→e1 + e2 + + en = k e1 e2 en g e1 + e2 + + en = k ←→ ∗, , ∗ | ∗, , ∗ | | ∗, , ∗ e1 e2 en (n − 1) − bars and k − stars f and g are one-to-one correspondences Definition 19 Let n, k ∈ N The symbol ( nk ) denotes the number of multisets with cardinality equal to k whose elements belong to an n-element set such as {1,2, ,n} Example 25 We can calculate (a) ( n1 ) =n (b) ( 23 ) =4 (c) ( k1 )=1 13 Properties (i) ( nk ) = n+k−1 k , where n k = n! k!(n − k)! n (ii) ( nk )=( n−1 k )+( k−1 ) (iii) Let n be a positive integer and let k be a nonnegative integer The equation e1 + e2 + + en = k has ( nk ) solutions for which e1 , e2 , , en are nonnegative integers Counting list and set summary Let S be a set with n elements (1) How many subsets of S are possible? (2) How many k-element lists are possible where (i) element repetition is not allowed? (ii) element repetition is allowed? (3) How many k-element subsets of S are possible where (i) element repetition is not allowed? (ii) element repetition is allowed? Counting functions Problem Let A and B be finite sets How many functions from A to B? Suppose A = {1, 2, , a} and B = {1, 2, , b} A function f : A −→ B, then f can be written as f = {(1, ∗), (2, ∗), , (a, ∗)}, where ∗ ∈ B (i) How many choices for ∗ entries? (ii) How many f ’s are one-to-one? (iii) How many f ’s are onto? 14 4.1 Pigeonhole principle Let A and B be finite sets and let f : A → B (i) If |A| > |B|, then f is not one-to-one (ii) If |A| < |B|, then f is not onto Example 26 How many functions from {1, 2, 3} to {a, b} are possible where (i) they are one-to-one? (ii) they are onto? 4.2 Theorem of counting functions Theorem Let A and B be finite sets with |A| = a and |B| = b (i) The number of functions from A to B is ba (ii) If a ≤ b, the number of one-to-one functions f : A → B is (b)a = b(b − 1) (b − a + 1) = b! (b − a)! If a > b, the number of functions is zero (iii) If a ≥ b, The number of onto functions f : A → B is b (−1)j j=0 b (b − j)a j (iv) If a = b, the number of bijiection functions f : A → B is a! If a = b, the number of such functions is zero Example 27 Given distinct lattice points in the plane, at least one of the line segments determined by these points has a lattice point as its midpoint 4.3 Permutations Definition 20 Let A be a set A permutation on A is a bijection from A to itself Example 28 Let A = {1, 2, 3, 4, 5} and f : A → A by f = {(1, 2), (2, 4), (3, 5), (4, 3), (5, 1) is a permutation on A 15 f is expressed in a matrix: f= 5 Properties Let Sn be the set of all permutations on {1, 2, , n} (i) |Sn | = n! (ii) ∀π, σ ∈ Sn , π ◦ σ ∈ Sn (iii) ∀π ∈ Sn , π −1 ∈ Sn and π ◦ π −1 = ι where ι(i) = i ∀i = 1, 2, , n 4.4 Cycle notation Definition 21 Let π ∈ Sn , a cycle by π is a sequence written as (x1 , x2 , , xk ) where xi s are pairwise different in {1, 2, , n} and xi+1 = π(xi ), π(xn ) = x1 Example 29 Let π = 5 Find cycles of π? π has two cycle: (1) and (2,3,4,5) π is expressed by using cycle notation as π = (1)(2, 3, 4, 5) or π = (2, 3, 4, 5) Composition symbol ◦ is omitted 4.5 Transposition Definition 22 A permutation τ ∈ Sn is called a transposition provided τ has only a cycle of the form (i, j), where ≤ i < j ≤ n Example 30 π = , ⇒ π = (1, 2) is a transposition Properties If τ ∈ Sn is a transposition, then τ ◦ τ = ι and τ = τ −1 4.6 Permutation decomposition Theorem Let π be any permutation on a finite set Then π can be expressed as the composition of transpositions defined on that set 16 Example 31 Let σ = 5 σ = (1, 4, 3, 2, 5) = (1, 5) ◦ (1, 2) ◦ (1, 3) ◦ (1, 4) or σ = (1, 4) ◦ (4, 3) ◦ (2, 3) ◦ (2, 5) any way else? Properties Let π be any permutation on a finite set and π is decomposed into transpositions as π = τ1 ◦ τ2 ◦ ◦ τa (i) π −1 = τa ◦ τa−1 ◦ ◦ τ1 (ii) If π is decomposed by another way such as π = σ1 ◦ σ2 ◦ ◦ σb , then a and b have the same parity Definition 23 Let π be a permutation on a finite set i) π is called even provided it can be written as the composition of an even number of transpositions ii) π is called odd provided it can be written as the composition of an odd number of transpositions Example 32 Let σ= 5 Is σ even or odd? 4.7 Exercises For each of the following relations defined on the set {1, 2, 3, 4, 5}, determine whether the relation is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive a) R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} b) R = {(1, 2), (2, 3), (3, 4), (4, 5)} c) R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)} d) R = {(1, 1), (1, 2), (2, 1), (3, 4), (4, 3)} 17 e) R = {1, 2, 3, 4, 5} × {1, 2, 3, 4, 5} For each of the following relations on the set of human beings, please determine whether the relation is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive a) has-the-same-last-name-as b) is-the-child-of c) has-the-same-parents-as (i.e., same mother and father) d) is-married-to e) is-an-ancestor-of f) R = {(a, b)|a, b ∈ N, a|b} (a|b ⇔ b a) g) R = {(a, b)|a, b ∈ Z, a − b 10} Let us say that two integers are near one another provided the absolute value of their difference is or smaller (i.e., the numbers are at most apart) For example, is near to 5, 10 is near to 9, but is not near to Let R stand for this is-near-to relation Please the following: a) Write down R as a set of ordered pairs Your answer should look like this: R = {(x, y) : } b) Prove or disprove: R is reflexive c) Prove or disprove: R is irreflexive d) Prove or disprove: R is symmetric e) Prove or disprove: R is antisymmetric f) Prove or disprove: R is transitive For each of the following relations, find R−1 a) R = {(1, 2), (2, 3), (3, 4)} b) R = {(1, 1), (2, 1), (3, 1)} c) R = {(x, y) : x, y ∈ Z, x − y = 1} d) R = {(x, y) : x, y ∈ N, x|y} e) R = {(x, y) : x, y ∈ Z, xy > 0} Which of the following are equivalence relations? a R = {(1, 2), (2, 3), (3, 1)} on set {1,2,3} b | on Z (a|b ⇔ b is divisible by a) c ≤ on Z 18 d “Is-an-anagram-of” on the set of English words (For example, STOP is an anagram of POTS because we can form one from the other by rearranging its letters.) For each of the following equivalence relations, determine the number of equivalence classes that relation has a Congruence modulo (for the integers) b Lives-in-the-same-province-as (for residents of Vietnam) Write out all the possible two-letter “words” one can make using only the vowels A, E, I, O, and U How many of these have no repeated letter? Airports have names, but they also have three-letter codes E.g., the Hanoi airport code is HAN, and the code FRA is for the Frankfurt airport in Germany How many different airport codes are possible? A bit string is a list of 0s and 1s How many length-k bit strings can be made? 10 How many different anagrams (including nonsensical words) can be made from each of the following? 1) DISCRETE 2) MATHEMATICS 3) INFORMATICS 11 Twelve people join hands for a circle dance In how many ways can they this? 12 You wish to make a necklace with 20 different beads In how many different ways can you this? 13 One hundred people are to be divided into ten discussion groups with ten people in each group In how many ways can this be done? 14 Let A ={1,2, , 100} How many 10-element subsets of A consist of only odd numbers? 15 How many partitions, with exactly two parts, can be made of the set {1,2,3,4}? Answer the same question for the set {1,2,3, ,n} 16 Let A be a 100-element set Which is greater: the number of partitions of A into 20 parts of size or the number of partitions of A into parts of size 20? 19 17 Ten married couples are seated around a large circular table In how many different ways can they this, assuming husbands and wives sit next to one another? Please note that if everyone moves one (or more) places to the left, the arrangement is not considered to be different 18 Let A be the set {1,2,3,4,5} a) How many ways to partition A into two parts are possible? b) How many ways to partition A into three parts are possible? c) How many ways to partition A into four parts are possible? d) How many possible equivalence relations are there on set A? 19 What multiset is encoded by the stars-and-bars notation *|||***? 20 Please calculate a) ( 24 ) b) ( 53 ) 21 Let n and k be positive integers Consider this equation: x1 + x2 + + xn = k a) How many solutions are there if the variables xi must be nonnegative integers? b) How many solutions are there if the variables xi must be positive integers? c) How many solutions are there if the variables xi may only take the values or 1? 22 Let n ∈ N Then there exist positive integers a and b, with a = b, such as na − nb is divisible by 10 23 In a meeting of n members, know that anyone shakes hand with at least one other person Prove that there exist two people who have the same number of hand shakes with others in the meeting 24 Let f = {(1, 2), (2, 3), (3, 4)} and g = {(2, 1), (3, 1), (4, 2)} be functions Please answer the following: a) What is f(2)? b) What is f(4)? c) What is dom(f)? d) What is im(f)? e) What is f −1 (1)? f) Prove that g −1 is not a function 20 g) What is g ◦ f ? h) What is f ◦ g? 25 Let A={1,2,3,4} and B= {1,2,3} a) How many one-to-one functions from A to B are possible? b) How many onto functions from A to B are possible? c) How many function f from A to B are possible where f (1) ≥ 1? 26 Let A={1,2,3,4} and B= {1,2,3,4} a) How many functions from A to B are possible? b) How many one-to-one functions from A to B are possible? c) How many function f from A to B are possible where f(1) is even? 27 Let f and g be permutations on set {1,2,3,4,5,6}, they are defined as follows: f= 6 g= 6 Decompose f into transpositions a) Find f −1 and g −1 b) Calculate f ◦ g and g ◦ f c) Express f and g by using cycle notation d) Decompose f and g into transpositions e) Are f and g even or odd? 21 ... Definition 13 Let A, B, and C be sets and let f : A → B and g : B → C Composition of g and f , denoted g ◦ f , defined by (g ◦ f )(a) = g[f (a)] Example 15 Let f : Z → Z by f (x) = x2 +1 and g : Z → Z... Theorem of counting functions Theorem Let A and B be finite sets with |A| = a and |B| = b (i) The number of functions from A to B is ba (ii) If a ≤ b, the number of one-to-one functions f : A →... (ii) element repetition is allowed? Counting functions Problem Let A and B be finite sets How many functions from A to B? Suppose A = {1, 2, , a} and B = {1, 2, , b} A function f : A −→ B,