Relations

Cartesian Product The Cartesian Product of A and B is defined as A X B = { (a,b)|a Î A and b Î B}. The Cartesian Product of n sets A1,A2 , A3 ,…………… An is defined as A1 X A2 X A3 X ……………….. X An = { (a1,a2,………an)|ai Î Ai ,i=1,2....n}. The Expression (a1,a2,………an) is called an ordered n-tuple. Example: Let A = { 0, 1, 2} and B = {a, b}. Then, A X B = {(0,a),(0,b),(1,a),(1,b),(2,a),(2,b)} A X A = { (0,0),(0,1),(0,2), (1,0),(1,1),(1,2), (2,0),(2,1),(2,2)}

Relations As Sets Computing is the science (and art) of processing information. A major concern is the relationship between different pieces of information. Databases. Examples of relations : John has savings account number 333. The File menu contains the exit option. The root directory is the parent of the directory tree. A binary relation can be represented as a set of pairs. N-ary relations are simply an extension of this. N-ary is mathematics speak for “any number of elements”. For example, a trinary relation could be written siblings(tim, dave, sharon) Databases are the second most import subject in Computing (after programming). Relational languages are all based on the Unification Algorithm invented by Alan Robinson. A Machine Orientated Logic Based On The Resolution Principle A.J. Robinson Journal Of The ACM Volume 22 pp22-41 Without Robinson’s Unification Algorithm Relational Databases wouldn’t work.

Relations in Computing Something like 80% of the software systems in existence involve databases. The vast majority of the databases in existence are based on relations : Relational Databases. There is a whole programming paradigm based on relations : the Relational programming languages. (PROLOG) Relational database query languages (SQL etc.) are Relational languages.

Source And Target Assume we have a set : Transport = { Car, Bike, Horse, Plane } The owns relation is a subset of People x Transport. owns = { (Dick, Bike), (Joe, Car), (Mary, Car) } In this case People is the source set of the relation owns and Transport is the target set of the relation owns. Note that not all the elements of either source or target sets needs to occur in the actual relation. The relation may be partial. (x  Transport  (Stan, x)  owns) x  Transport  (Stan, x)  owns In fact, it turns out that most useful relations are partial.

Domain And Range The set of elements which occur as the first of the pairs in a relation is called the domain (dom) of the relation. The set of elements which occur as the second of the pairs in a relation is called the range (ran) of the relation. dom(owns) = { Dick, Joe, Mary } ran(owns) = { Bike, Car } As a pretty picture : Yet again we have different names for the same things. domain and input are synonyms. range and output are synonyms. Also note not that domain  source range  target Dick owns Bike owns (Dick, Bike) (Dick, Bike)  owns Bike Mary Dick Car Stan Plane Joe

A relation is a set of ordered pairs. The domain is the set of all x values in the relation domain = {-1,0,2,4,9} These are the x values written in a set from smallest to largest This is a relation {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} These are the y values written in a set from smallest to largest range = {-6,-2,3,5,9} The range is the set of all y values in the relation Relations are the theoretical basis for Relational programming languages such as Prolog and SQL.

A relation assigns the x’s with y’s 1 2 2 4 3 6 4 8 10 5 Domain (set of all x’s) Range (set of all y’s) This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)}

Relational Inverse The inverse of a relation is obtained by swapping round the components of each pair. R~ = { (1,0), (2,0), (2,1), (5,3) } S~ = { (2,1), (5,3) } Mathematically, we can write S~  { (x,y)  S  (y,x) } A good example of the use of a relational inverse is the child relation. child = parent~ Actually, in most Relational programming languages you get the inverse of a relation “for free”. The query parent(X, dave) would produce the set X = {jim, theresa } parent(theresa, Y) Y = {tim, dave, sharon }

Relational Composition The composition of R with S is equal to the set of all those ordered pairs (x, z) such that we can find at least one value, say y, such that xRy and ySz. That is R ; S  { (x,y)  R, (y, z)  S  (x, z) } As usual, an alternative notation is sometimes used. Instead of R ; S we write S o R to mean the same thing. So R ; S = { (0,1), (1, 1) }. 1 3 2 5 A good example of the use of a relational composition is the grandparent relation. grandparent = parent ; parent If we add my brother’s children to our family database parent(tim, james). parent(tim, thomas). the query grandparent(theresa, Z) would produce the set Z = {james, thomas } This happens because the parent database already includes the following record : parent(theresa, tim).

Relational Override This is essentially an update operation. The override of R by S is denoted R  S by adding to S all those ordered pairs from R whose first component is not in the domain of S. R  S  R  S \ { (x,y)  R  x  dom S } This sounds a lot more complex than it is. Lets have an example. R = { (1,2), (3,4), (2,3) } S = { (1,6), (4,5), (2,7) } R  S = { (1,6), (4,5), (2,7), (3,4) } A good way to think of this is that R is the master database and S is a set of new records to be added, some of which update the data in already existing records. We used relational override on our parent relation on the last slide to add james and thomas. Another good example is the update that happens every academic year when this institution updates SITS (Student IT System) which entries for all the new students and the module results for all the continuing students. The new students’ records are added to the database. The continuing students’ records are amended to show their new results.

Diagrammatic Representations Relations are often best understood when represented by a pretty picture. A directed graph or digraph. R = { (1,2), (2,1), (2,2), (2,4), (2,5), (3,1), (3,5), (4,5), (5,5) } The following diagraph represents R : 2 1 3 5 4

Diagrammatic Representations II Given R = { (1,2), (4,2), (4,3), (1,3), (4,1), (3,2), (3,3), (4,5) } The following diagraph represents R : 4 3 2 1

Types Of Relations An homogeneous relation is a binary relation in which the source and target sets are identical. R  Names  Names Useful for things like A family tree (parent, ancestor, married to etc.) Traveling problems (bus, rail and road connections etc.) An identity relation is a binary relation in which every element of a set is related to itself and only to itself. Ix = {x  X  (x,x) } For example if X = { 0, 1, 2 } then Ix = { (0,0), (1,1), (2,2) } The parent and grandparent relations are homogenous relations on the set of People. The most famous traveling problem is called the Traveling Salesman Problem (TSP). Given a set of cities find the shortest route to visit each city once (and only once). Sounds simple. It’s not. The TSP is actually the simplest example of an NP-Complete problem. Basically, that means that it takes a long time to solve. NP stands for Nondeterministic Polynomial. In English, very, very slow. Another example of an NP-complete problem is RSA encryption. Which makes Internet security very, very slow. RSA is a public / private (asymmetric) key encryption algorithm. RSA is named after its supposed inventors, Ron Rivest, Adi Shamir and Leonard Adelman. The story is that public / private key encryption was first invented in the 1960s at GCHQ. Obviously, the spooks didn’t tell anyone.

Types Of Homogenous Relations Can R = R~? Obviously, only for homogeneous relations. Such a relation is called symmetric. Given a relation R on the set X, for a symmetric relation : x,y  X  (x,y)  R  (y,x)  R Reflexive relations : every element of the domain is related to itself. Ix  R Not quite the same as a identity relation. There may be other pairs as well. For example : R = { (1,1), (2,2), (1,5), (2,3) }

Types Of Homogenous Relations II If whenever a is related to b and b is related to c then a is related to c then the relation is transitive. R ; R  R If a relation is reflexive, symmetric and transitive then it is called an equivalence relation. Given the relation R on the set { 1, 2, 3 } R = { (1,1), (1,2), (2,1), (2,2), (3,3) } Is this relation : Reflexive? Symmetric? Transitive? An equivalence relation? R is reflexive, symmetric and transitive so it is an equivalence relation. In our family database we could define a transitive relation ancestor. ancestor = parent  parent ; ancestor In English, one of your ancestors is either your parent of the parent of another of your ancestors.

Reflexive xRx for all x Î A Symmetric If xRy, then yRx, for all x,y Î A Transitive If xRy and yRz, then xRz, for all x,y,z Î A Equivalence relation: Let R be a relation on A. R is an equivalence relation on A if the following conditions are satisfied: 1. xRx for all x Î A ( R is reflexive) 2. If xRy, then yRx, for all x,y Î A ( R is symmetric) 3. If xRy and yRz, then xRz, for all x,y,z Î A ( R is transitive)

Example: Prove R is an equivalence relation Example: Prove R is an equivalence relation. Let N be the set of natural numbers,that is, N = {1,2,3……..}. Define a relation R in N as follows: R = {(x,y)| x,y Î N and x + y is even}. Example :Let A = { 1,2,3,4}, and let R = { (1,3),(4,2),(2,4),(2,3),(3,1)}. Is it R is equivalence or not Antisymmetric relation: A relation R on a set A is called an antisymmetric relation if (a,b) Î R and (b,a) Î R implies a = b. Example: R = { (1,3),(4,2),(2,4),(2,3),(3,1)}. R is not antisymmetric because (1,3) Î R and (3,1) Î R, but certainly 1 ≠ 3.

Problems Let A = {1,2,3,4,5,6}. Construct the relation R on A for the following cases. a) R = {(j,k) | j divides k} b) R = {(j,k) | j is a multiple of k} c) R = {(j,k) | (j – k)2 Î A} d) R = {(j,k) | j/k is a prime} 2. Let R be the relation from A = {1,2,3,4,5} to B = {1,3,5} which is defined by “ x is less than y.” Write R as a set of ordered pairs. 3. Let R be the relation in the natural numbers N = {1,2,3,…..} defined by “x + 2y =10”, that is ,let R = {(x,y) | x Î N, y Î N, x + 2y = 10}. Find a) the domain and range of R b) R-1