f2006lecNotesWeek09 - 24 24.1 Relations Introduction...

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24 Relations 24.1 Introduction consider sets A = { Mon, Ott, Tor, Van } , B = { BC, ON, QC } recall: the Cartesian product A × B = { ( a, b ) | a A, b B } = { (Mon, BC), (Mon, ON), ... , (Van, ON), (Van,QC) } . suppose we are only interested in the ordered pairs ( a, b ) where city a is in province b . then we have a subset of A × B R = { (Mon, QC), (Ott, ON), (Tor, ON), (Van, BC) } R is a relation from the cities to the provinces. a binary relation R from a set A to a set B is a subset of A × B . if ( a, b ) R , then we say a is related to b by R , denoted aRb . if ( a, b ) ±∈ R , then we say a is not related to b R , denoted a ± Rb . in previous example, R = { ( a, b ) A × B | city a is in province b } (Tor, ON) R Tor R ON Ott, BC) ±∈ R Ott ± R BC an n -ary relation on the sets A 1 ,A 2 ,...,A n is a subset of A 1 × A 2 ×···× A n we will not be looking at these in our course when we talk about relations, we will mean binary relations 24.2 Graphical representation of a relation suppose R is a relation from A to B draw A, B as regions, elements of A, B as dots if , draw an arrow from a to b display cities, provinces relation using this 24.3 Functions and relations a function F from A to B is a relation from A to B such that x A y B ( xFy ) x A y B z B ( xF y ) ( xFz ) ( y = z ) use arrow diagrams to display examples of functions and non-functions 24.4 Inverses of relations given a relation R from A to B , the inverse relation R - 1 from B to A is: R - 1 = { ( x, y ) B × A | ( y,x ) R } = { ( ) B × A | ( x, y ) R } draw arrow diagrams of a simple relation and its inverse 24.5 Relations on a set a relation on a set A is a binary relation from A to A . e.g. A = { 1 , 2 , 3 , 4 } , R = { (1 , 1) , (2 , 1) , (2 , 2) , (3 , 1) , (3 , 3) , (4 , 1) , (4 , 2) , (4 , 4) } i.e. xRy x is a multiple of y 52
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24.6 Graphical representation of a relation on a set we modify our previous technique as follows: since we have only one set, A , we do not need two regions we do not draw the boundary of the single region use dots and arrows as before: aRb draw an arrow from a to b the result is called a directed graph or digraph e.g. draw the digraph representation of R in the previous example 24.7 Some useful classiFcations of relations
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f2006lecNotesWeek09 - 24 24.1 Relations Introduction...

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