Understanding Functions and Relations: Concepts, Products, and Uses, Lecture notes of Computer Science

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which can be relations or functions make sets of pairs Common things one can do with sets: Functions and relations Friday, September 24, 2010 2:51 PM 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 Saturday, September 25, 2010 4:00 PM 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 Friday, September 24, 2010 2:57 PM 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 Friday, September 24, 2010 3:01 PM 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 Friday, September 24, 2010 3:04 PM 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 Friday, September 24, 2010 4:01 PM Docsity.com

A function can be recognized by noting that only one arrow points away from each element of the domain. Recognizing a function Friday, September 24, 2010 4:04 PM Docsity.com

one-to-one (or injective ): each value in D maps to a different value in R. onto (or surjective ): all values in R are part of some pair. one-to-one and onto (or bijective ): there is a one-to-one correspondence between all of D and all of R. partial : only some x in D have f(x) defined. A function can be Kinds of functions Friday, September 24, 2010 4:06 PM Docsity.com

Onto function Friday, September 24, 2010 4:13 PM Docsity.com

If D is a set, f(D) = { f(x) | x ϵ D}. Thus we have another equivalent definition of surjective: f:D→R is surjective ↔ f(D)=R. Functions of a set Saturday, September 25, 2010 3:42 PM Docsity.com

A bijective function f:A→B has an inverse function f

• 1 : B→A , where f(a)=b ↔ f
• 1 (b)=a. Inverse functions Saturday, September 25, 2010 3:37 PM Docsity.com

The composition of two functions f:D→E and g:E→F is a set g f:D→F = { (x,z) | x ϵ D z=g(f(x)) }. Composing functions Saturday, September 25, 2010 4:07 PM Docsity.com

An array is a function from integers to values. foreach $x (0,1,2,3) { $square[$x]=$x*$x; } results in @square=(0,1,4,9); An array is a function Saturday, September 25, 2010 4:12 PM Docsity.com

A hash is a function from scalars to scalars. Example: %h = ("01201" => "Harry", "05560" => "Sue", "07707" => "Mark"); domain of %h is the set {"01201", "05560", "07707"}. range of %h is the set {"Harry", "Sue", "Mark"}. Each element of the domain is a unique key. A hash is a function Saturday, September 25, 2010 4:14 PM Docsity.com