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 numbers. But relations also can hold not just between numbers, but also between other kinds of mathematical objects: for example, i) for two triangles? EFG and? HIJ, (congruence) is a relation that might hold between them; another relation is µ ÐsimilarityÑÞ ii) for two lines j" and j in the plane, ² Ðparallel to) is a relation that might hold between them à another example is ¼ (perpendicular). So what is a relation? It appears to be a very general concept. Can we say something more precise than A relation is, well, you know, like Ÿ or ²? To answer this question, we start with the idea of an ordered pair. Ordered Pairs and Products of Sets Informally, an ordered pair Ð+ß,Ñ consists of objects + and,, and is characterized by having them in order : there is a first object +ß and a second object ß,Þ We consider two ordered pairs to be the same Ðand write Ð+ß,ÑœÐ-ß.ÑÑ +œ- and,œ.þ Therefore the ordered pairs Ð"ß Ñ and Ðß "Ñ are derent. ( For example, if you think of these ordered pairs as being the coordinates of points in the plane, then certainly Ð"ß Ñ and Ðß "Ñ represent derent points.) In the same way, we can think of ordered triples Ð+ß,ß-Ñ or even ordered 8-tuples Ð+ " ß + ß ÞÞÞß + 8 Ñ. We say that two ordered 8-tuples Ð+ " ß + ß ÞÞÞß + 8 Ñ and Ð, " ß, ß ÞÞÞß, 8Ñ are equal + œ, for each 3 œ "ß ÞÞÞß 8Þ 3 3 This informal way of thinking about ordered pairs, ordered triples,..., ordered 8-tuples is usually good enough for doing everyday mathematics. But notice that we have not really defined an ordered pair À we just said that Ð+ß,Ñ is characterized by... a certain property. We have said nothing like an ordered pair Ð+ß,Ñ is....
As always, mathematicians don't really care (except if they become philosophers) what an ordered pair really is all they want is a precise definition of something that acts just like an ordered pair, something that has all the relevant characteristics of an ordered pair. (described above). Moreover, if everything in mathematics is a set, then an ordered pair should be defined as some set. So we want to write a definition of the form Ð+ß,Ñ œ some set. We certainly can't define Ð+ß,Ñ œ Ö+ß,, since, for example, Ð"ß Ñ Á Ðß "Ñ while Ö"ß œ Öß " Þ The set Ö"ß œ Öß " doesn't distinguish between a first element and a second element. It doesn't really matter what particular set we define ordered pair Ð+ß,Ñ to be, as long as the set behaves the way an ordered pair should behave: that Ð+ß,Ñ œ Ð-ß.Ñ + œ - and,œ.þ One possible definition, devised by the Polish mathematician K. Kuratowski near th the beginning of the 20 century, is the one the most mathematicians use when they need a formal definition of an ordered pair. Definition Ð+ß,Ñ œ Ö Ö+ ß Ö+ß,. We call Ð+ß,Ñ an ordered pair. This formal definition gives us a set Ð+ß,Ñ and the next theorem says that it has the characteristic property of an ordered pair (and therefore we can live with this definition Ð+ß,Ñ as this particular set). Theorem Ð+ß,Ñ œ Ð-ß.Ñ + œ, and - œ.. Proof If + œ, and - œ., then Ö Ö+ ß Ö+ß, œ Ö Ö- ß Ö-ß.. Conversely, we need to show that if these two sets are equal, then +œ- and,œ.. This is left as an exercise. (This is not dicult. But in the argument you do have to consider the possibility that +œ,, in which case the set on the left reduces to Ö Ö+ ß Ö+ß, œ Ö Ö+ ß Ö+ß + œ Ö Ö+ ß Ö+ œ Ö Ö+. ñ We can then make a formal definition of an ordered triple as an ordered pair whose first member is itself an ordered pair: Ð+ß,ß-ÑœÐÐ+ß,Ñß -Ñ Inductively, we can formally an Ð8 "Ñ-tuple based on the previous definition of an 8- tuple: Ð+ ß ÞÞÞß + ß + Ñ œ ÐÐ+ ß ÞÞÞß + Ñ ß + Ñ. " 8 8 " " 8 8 " Using the definition, it's easy to prove by induction that, for every 8, Ð+ " ß ÞÞÞß + 8 Ñ œ Ð, " ß ÞÞÞß, 8Ñ + 3 œ, 3 for each 3 œ "ß ÞÞÞß 8Þ
Notice that the formal definition means that ordered 8-tuples also are sets. For example, consider an ordered triple: Ð+ß,ß-ÑœÐÐ+ß,Ñß-ÑœÖÖÐ+ß,Ñ ß ÖÐ+ß,Ñß- œ Ö Ö ÖÖ+ ß Ö+ß, ß Ö ÖÖ+ ß Ö+ß, ß - In the same way, we can see that complicated one if 8 is large). Ð+ ß ÞÞÞß + ß + Ñ " 8 8 " is also just a set (a very Of course, mathematics would get very cumbersome if we had to think of ordered 8-tuples this way every time they come up. The point is that once we've convinced ourselves that an ordered pair (or 8-tuple) can be defined as a set, then we're satisfied that the idea of ordered 8-tuple rests on a precise definition in set theory and that the formally defined ordered 8-tuple acts just the way an informal ordered 8-tuple acts. Then we can relax and use ordered 8-tuples in the same way we always have. We are now ready to describe another way in which sets can be combined to form new sets. Definition If E and F are sets, then the product E F œ ÖÐ+ß,Ñ À + E and, F Þ More generally, if E" ß ÞÞÞß E8 are sets, then E " ÞÞÞ E8 œ ÖÐ+ " ß ÞÞÞß + 8Ñ À + 3 E3 for each 3 œ "ß ÞÞÞß 8 Note: If all the E3's are equal to the same set E, then we might write E instead of E E ÞÞÞ E Þ So, for example, we might write E instead of E EÞ " 8 8 Examples The BC -plane is the product set. 8... œ is the set of all ordered 8-tuples of real numbers called 8- dimensional Euclidean space. Ö! œ ÖÐ!ß,Ñ À, Þ In fact, Ö! is the C-axis in. If E œ Ö"ß and F œ Ö%ß &ß ', then E F œ ÖÐ"ß %Ñß Ð"ß &Ñß Ð"ß 'Ñß Ðß %Ñß Ðß &Ñß Ðß 'Ñ Notice that Ð"ß %Ñ E F but Ð"ß %Ñ Â F E. Usually E FÁF E. Can you invent and prove a simple theorem that states E FœF E...?
Abstract nonsense a crazy example (there are lots of these in the text) just to be sure you understand the idea: If E œ Ö ÖB ß gß $ and F œ ÖCß Ög, then E F œ Ö ÐÖB ß CÑß ÐÖB ß Ög Ñß Ðgß CÑß Ðgß Ög Ñß Ð$ß CÑß Ð$ß Ög Ñ There are a number of results in the textbook ( see Theorem 3.1 ) about how products of sets interact with unions and intersections. For example: E ÐF GÑ œ ÐE FÑ ÐE GÑÞ Proof ( As usual, to prove that two sets are equal we show that they have the same members. In this case, it's easy to write an proof, rather than showing separately, in two arguments, that the left side is a subset of the right side and vice-versa.) ÐBß CÑ E ÐF GÑ B Eand C F G B Eand C Fand C G ÐBß CÑ E F and ÐBß CÑ E G ÐBß CÑ ÐE FÑ ÐE GÑÞ ñ We can also defined products of more than two sets (inductively). For example E F G can be defined as a product of two sets ÐE FÑ G If a product of 8 sets has been defined, then we can define the product of 8 " sets by the formula E ÞÞÞ E E œðe ÞÞÞ E Ñ E Þ " 8 8 " " 8 8 " Notice that E F G œðe FÑ G œöðð+ß,ñß-ñà Ð+ß,Ñ E Fß - G œöð+ß,ß-ñà Eß, Fß- G ( because of how we defined an ordered triple Ð+ß,ß-Ñ.
Note: An Inconsistency with the Textbook On p. 132, the text states that E F G, ÐE FÑ G and E ÐF GÑare three derent sets. 1) If you just assume (as the text does) that you can write down ordered triples Ð+ß,ß-Ñ without defining them, then you can take the next step and write E F GœÖÐ+ß,ß-ÑÀ+ Eß, Fß- G (bottom of p. 132) Of course, you can also write ÐE FÑ G being itself a product of two sets): (a product of TWO sets, the first one ÐE FÑ G œ Ö Ð Ð+ß,Ñß -Ñ À Ð+ß,Ñ E Fß - G From the text's point of view, these are derent sets: the first one has ordered triples as members, the second one has as members ordered pairs (each with another ordered pair as its first object). 2) In class and in these notes, I took the trouble to actually DEFINE what an ordered triple is: we had already defined ordered pairs, so I defined order tiples in terms of ordered pairs: my definition was Ð+ß,ß-ÑœÐÐ+ß,Ñß-Ñ Under that definition, ÐE FÑ G œ Ö Ð Ð+ß,Ñß -Ñ À Ð+ß,Ñ E Fß - G œöð+ß,ß-ñà+ Eß, Fß - G œe F G But, with my definition, it is true that ÐE FÑ Gand E ÐF GÑare derent sets: members of the first set look like ÐÐ+ß,Ñß-ÑœÐ+ß,ß-Ñbut members of the second set look like Ð Ð+ß Ð,ß -ÑÑÞ 3) It turns out that in most mathematics where set products are actually USED, it doesn't matter whether you WORK WITH E F Gor ÐE FÑ G or E ÐF GÑ. Whenever you're doing elsewhere in math that requires use of sets, you can usually use either one of the three equally well as a tool. Therefore (as the textbook says at top of p. 133) the literal derence between these "threefold-products" can usually be safely ignored. But if you're doing elementary set theory per se, a derence DOES exist.
Relations To return to our original question: relations are things like Ÿß ß ß. If everything we in mathematics is a set, we should be able to give a formal definition like: a relation V is (... some set...), and the key to such a precise definition is to focus on the idea that a relation relates a pair of objects. Consider the following subset of À V œ ÖÐBß CÑ À b- ß B - œ C Þ Notice that the familiar relation +Ÿ, in could be defined as follows: +Ÿ, Ð+ß,Ñ VÞ For example " Ÿ %, and Ð"ß %Ñ V because % œ " Ð È$Ñ; "Ÿ", and (1,1) V because "œ"! œ"!. Note: the point of having - (rather then -) in the definition of V is that i) We don't want ÐBß CÑ in V if, say, B Ð Ñ œ CÞ ii) But we can't write the definition like VœÖÐBßCÑ À b- where B -œc and!ÿ- because: if we're trying to give a definition for +Ÿ, as meaning Ð+ß,Ñ Vß then it would be circular to use Ÿ as part of the definition of the set V. Using - in the definition of V lets us sneak in the idea we want via algebra, without referring to ŸÞ Since we are working here with numbers, we can even draw a picture of the set V : it consists of the set of all points in the BC-plane that sit on or above the graph of C œ B ( why?). A (partial) picture of the set V looks like:
Then +Ÿ, means the same thing as the ordered pair Ð+ß,Ñis a member of the set V(the shaded region. That is, the set V contains all the information about which numbers are and are not related by Ÿ. If we want to define Ÿ, formally and officially, as a set, then the choice is obvious: we can define Ÿ œ V œ ÖÐBß CÑ À b- ß B - œ C Then the relation Ÿ is the set VÞ Of course, we can the agree, just as notation, to write +Ÿ, to mean the same thing as Ð+ß,Ñ Ÿ. The example gives the idea we can use, in general, to formally define a relation as a set. Definition A relation V between E and F is a subset of E F. If Ð+ß,Ñ V, we also write +V,, meaning that + stands in the relation V to,. In the definition, the sets E and F can be derent. But sometimes E œ F (as in the example above, where EœFœ Ñ: in that case we also say that Vis a relation on the set E. According to the definition, we call Ÿ œ V (in the preceding example) a relation between and a relation between real numbers. In this case (and in general, when EœF in the definition) people sometimes say Vis a relation in E: but if you say this, remember that the set V is a subset of E EÞ