Section 4.2 Functions Announcement • Exam #3 has been rescheduled – Monday, April 4th – Chapters 3 and 4 – Practice sheet for Chapter 3 now posted. Activity 1 Let f: X → Y as shown by : • • • • What is the domain of f? What is the target of f? What is the range of f? What is f(a)? f(b)? f(c)? Review • What is a function? Definition of a function • A function f from a set X to a set Y, denote f: X → Y, is a relation from X, the domain to Y, the target (or co-domain) that satisfies two properties: 1. Every element in X is related to some element in Y 2. No element in X is related to more than one element in Y Consider the following • Let N = {Adams, Epp, Stevens, Rosen, Smedley} L = {A, B, C, D, F} g : N → L such that the mapping indicates the student’s grade in the class. Is g a function? Non-functions • What kind of situations would need to occur for this to not be a valid function. We sometimes write functions as a set of ordered pairs. g = {(Adams, A), (Epp, C), (Stevens, B), (Rosen, A), (Smedley, F)} Activity #2 • Represent f as a set of ordered pairs. Activity 3 Consider: L = {1,2,3,4,5} R = {A,B,C,D,F} Which of the following are functions from L to R? 1. {(1,A),(2,B),(3,C),(4,D),(5,F)} 2. {(2,A),(2,B),(3,C),(4,D),(5,F)} 3. {(1,A),(2,A),(3,B),(4,B),(5,B)} 4. {(1,5),(2,4),(3,3),(4,2),(5,1)} 5. {(2,A),(3,B),(4,B),(5,B)} Activity 4 • Find all of the functions from X = {a,b} to Y={u,v} • Find all of the functions from X = {a,b} to Y={u} • How many functions would exist from X = {a,b,c,d} to Y={u,v,x} • How does this relate back to Cartesian products of sets? Common Functions in CS • Floor function floor: R → Z floor(x) = the largest integer y such that y≤x Common Functions in CS • Ceiling function ceiling: R → Z ceiling(x) = the smallest integer y such that x≤y Common Functions in CS • Encoding and Decoding functions E: bitString → bitString D: bitString → bitString Common Functions in CS • Encoding and Decoding functions E(s) = the string obtained from s by replacing each bit of s by the same bit written three times D(t) = the string obtained from t by replacing each consecutive triple of three bits by a single copy of that bit Common Functions in CS • Encoding and Decoding functions E(s) = the string obtained from s by replacing each bit of s by the same bit written three times D(t) = the string obtained from t by replacing each consecutive triple of three bits by whichever bit is most frequent. Common Functions in CS • The Hamming Distance H: bitString,bitString → Z H(s,t) = the count of the number of positions in which s,t have different values