Example Let VœÖÐBßCÑ À b-, CœB - ( see the example above). Explain why

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Roger Floyd
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1 Definition If V is a relation from E to F, then a) the domain of V œ dom ÐVÑ œ Ö+ E À b, F such that Ð+ß,Ñ V b) the range of V œ ran( VÑ œ Ö, F À b+ E such that Ð+ß,Ñ V " c) the inverse relation of V œ V œ ÖÐ,ß +Ñ À Ð+ß,Ñ V. " " V is a relation from F to E (because V F E.) # Example Let VœÖÐBßCÑ À b-, CœB - ( see the example above). Explain why domðvñ œ ran ÐVÑ œ Þ " # In this case, V œ ÖÐCß BÑ À b-, C œ B - Þ Since (1,4) V and " " " " Ð"ß "Ñ V, we have Ð%ß "Ñ V and Ð"ß "Ñ V. We can write %V " and "V "Þ Of course, these particular relations V and V " are usually called by their more familiar " names Ÿ and, so we write " Ÿ % and % " rather than "V% and %V "Þ Every subset of œ # is a relation on, so there are infinitely many possible relations on. For example, V œ ÖÐ1ß #Ñß Ð/ß È# Ñß Ð"ß (Ñ Þ For this V, dom ÐVÑ œ Ö1ß /ß " and ran ÐVÑ œ Ö#ß È# ß (, so we can write 1V#ß /V È # ß and "V(. But this particular relation V is of no interest to anybody, so it has never been given a familiar name (unlike the relation Vœ Ÿ). Most of the infinitely many possible relations on are uninteresting. Exercise Define the familiar relations and as sets (avoiding, of course, the use of and in the definitions). Example Let E œ Ö ÖB ß gß $ and F œ ÖCß Ög ß Dß A and V œ Ö ÐÖB ß CÑß Ðgß CÑß ÐÖB ß Ög Ñ E F. V is a relation from E to F. We can write, for example: ÖB VC and $VC Î. dom ÐVÑ œ Ö ÖB ß g E (but dom ÐVÑ Á E) ran ÐVÑ œ ÖCß Ög F (but ran ÐVÑ Á F) V œ Ö ÐCß ÖB Ñß ÐCß gñß ÐÖg ß ÖB Ñ, so, for example, CV gþ Example Suppose 7ß 8 = and that E has 7 elements and F has 8 elements. Then 78 E F has 78 elements (ordered pairs), and a set with 78 elements has # subsets. 78 Therefore there are exactly # different relations from Eto F.

2 Example If V œ ÖÐ7ß 8Ñ À Ðb5 Ñ 8 œ 57. V is a relation on that has a more common name: lœöð7ß8ñ ÀÐb5 Ñ8œ57 We write 7l8 iff Ð7ß 8Ñ lþ For example, we write #l% because Ð#ß %Ñ lþ Note: This set is the relation on. The divides relation on is defined by a different set ÖÐ7ß8Ñ À Ð b5 Ñ8œ57. But it's customary to use the same symbol, l, for this relation. Context usually tells us whether l refers to the division relation in or the division relation in. We do the same thing with other relation symbols for example, Ÿ ( on ) was defined above as a certain set. We use the same symbol, Ÿ for the corresponding relation in, say, =. Example We can define a relation 0 on by # 0 œ ÖÐBß CÑ À C œ B. The plot of this set of points in is a familiar parabola: the graph of the function CœB #. This suggests that functions can perhaps be thought of as special kinds of relations ( we will give a careful definition of function later ). # Dom Ð0Ñ œ RanÐ0Ñ œ Ò!ß _Ñ Ð#ß %Ñ 0 and Ð #ß %Ñ 0ß so we write #0% and #0%Þ However in this situation, it's more customary to use function notation and write %œ0ð#ñand %œ0ð #ÑÞ Both " " Ð%ß#Ñ 0 and Ð%ß #Ñ 0 Þ (When we have defined functions, it will turn out that although this particular perfectly good relation, it is not a function.) 0 " is a Can you guess what additional property or properties a relation should have to be called a function? We can also introduce the more general idea of composition of relations here, although composition of relations is mostly used when the relations are actually functions. Definition Suppose V is a relation from E to F and that W is a relation from F to CÞ We define the composition relation W Vfrom Eto G by W V œ ÖÐ+ß -Ñ À b, F such that Ð+ß,Ñ V and Ð,ß -Ñ W.

3 Examples 1. Suppose E œ Ö"ß #ß $ ß F œ Ö#ß $ß %ß & ß G œ Ö+ß,ß - and let V œ ÖÐ"ß $Ñß Ð"ß %Ñß Ð#ß &Ñ ß W œ ÖÐ$ß +Ñß Ð$ß,Ñß Ð&ß +Ñß Ð&ß,Ñ Then W V œ ÖÐ"ß +Ñß Ð"ß,Ñß Ð#ß +Ñß Ð#ß,Ñ (a subset of E G). dom ÐW VÑ œ Ö"ß #, a subset of E (and in this case, Á E) ran ÐW VÑ œ Ö+ß,, a subset of G (and in this case, Á G). #. Define relations V and W on by V œ ÖÐBß CÑ À B and C œ sin B and WœÖÐBßCÑÀB, B! and CœÈB. We could also write these as V œ ÖÐBß sin BÑ À B and W œ ÖÐBß È B Ñ À B, B!. Then W VœÖÐBßÈ sin B ÑÀB and sin B! and V W œ Ö ÐBß sin ÐÈ B Ñ Ñ À B and B! Þ Usually, V WÁW VÞ Most of the relations that exist on a set are not very interesting or useful. Relations become more interesting when then have some special additional properties. Definition Let V be a relation on a set E (that is, V E E). V is called a) reflexive iff a+ Eß +V+ b) symmetric iff a+ E a, E Ð+V, Ê,V+Ñ c) transitive iff a+ E a, E a- E Ð +V,,V-Ñ Ê +Vd) antisymmetric iff a+ E a, E Ð+V,,V+Ñ Ê Ð+ œ,ñ

4 Example Check that each Yes or No in the table is correct. Ÿ l (on ) (on ) (on ) (on c(a) Reflexive Symmetric Transitive Antisymmetric Y N Y Y N N Y Y Y N Y N Y N Y Y Comments about a couple of the table entries might be helpful. The divides relation l on is antisymmetric, but not on. Why is an antisymmetric relation on? Consider the negation of the definition: V is not antisymmetric means µ Ð a+ a, Ð+V,,V+Ñ Ê Ð+ œ,ñ Ñ b+ b, µ Ð+V,,V+Ñ Ê Ð+ œ,ñ Ñ b+ b, Ð +V,,V+ Ð+ Á,Ñ Ñ which is equivalent to which is equivalent to For to be not antisymmetric there would have to exist real numbers +ß, such that Ð+,Ñ and Ð, +Ñ and Ð+ Á,Ñ. This is impossible. Definition A relation Von a set E is called an equivalence relation iff V is reflexive, symmetric, and transitive. Symbols like,, and µ are often used to denote equivalence relations. None of the relations listed in the table above is an equivalence relation. In everyday language, when we say that two things are equivalent, the meaning is not that they are necessarily the same thing, but that for certain purposes they can be treated as if they were the same. For example, we might say that brothers and sisters are equivalent under Missouri inheritance law. Of course that is not to say there is no difference between brothers and sisters, but rather that for purposes of inheritance law, they can't be distinguished from each other: gender doesn't matter. In mathematics, an equivalence relation has roughly the same purpose. For example, the trig function sine can't distinguish between Band B # 1 À since sin B œ sin ÐB # 1Ñ, the two numbers are equivalent for the purpose of computing values of the sine function. Example Let _ œö6à 6is a straight line in the BC-plane whose equation has the form C œ %B, For two points TßU in the plane, #, define T U iff b6 _ such that TU, are both on the line 6

5 If we use coordinates and write TœÐBßCÑ, UœÐDßAÑ, then the relation can be written as # # œ ÖÐ ÐBß CÑß ÐDß AÑ Ñ À ÞÞÞ. What is the condition...? Check that is an equivalence relation on #. For a particular point T in the plane, describe the points that are in the set ÒTÓœÖU # À T U? ( This set is called the equivalence class of T Þ)

Relations. Relations occur all the time in mathematics. For example, there are many relations between # and % À

Relations. Relations occur all the time in mathematics. For example, there are many relations between # and % À Relations Relations occur all the time in mathematics. For example, there are many relations between and % À Ÿ% % Á% l% Ÿ,, Á ß and l are examples of relations which might or might not hold between two