Functions - Biological and Cognitive Sciences - Lecture Notes, Study notes of Biogenetics and Computers

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which can be relations or functions make sets of pairs Common things one can do with sets: Functions and relations Docsity.com

You are probably used to functions such as f(x)=x 2 . For every x, there is a y, such that y=x 2 . Let's generalize this concept. Familiar and unfamiliar concepts of function Docsity.com

A relation between two sets A, B is some subset of A x B. Example: A = { "Fred", "George" }, B = {"cars" ,"trucks" } A x B = { R = { ("Fred", "cars"), ("Fred", "cars"), ("George", "cars"), ("Fred", "trucks"), ("George", "trucks") ("George", "cars"), } ("George", "trucks") } Relations Docsity.com

E.g., interpret (x,y) as "x likes y" in the previous example. Defining the relationships between two sets of things E.g., interpret (x,y) as "x is a friend of y". Defining friends in a set of users Uses of relations: Using relations Docsity.com

A function is a relation in which for every x, there is a unique y such that (x,y). This can be written as: (x,y) ϵ R (x,z) ϵ R → y=z More generally: "There is a unique q such that P(q)" can be written as P(q) P( r) - > q=r Functions Docsity.com

domain (things that can go on the left) range (things that can go on the right) A function has a "f is a function from domain D to range R" is often notated as f: D → R This is not implication; it is mapping. For x in D, the corresponding element of R is f(x). Thus the function is a set of pairs f=(x, f(x)). Thus f is a set, and f(x) is a lookup of the pair (x,y) ϵ f Function concepts Docsity.com

A function can be recognized by noting that only one arrow points away from each element of the domain. Recognizing a function Docsity.com