Return-Path: Message-Id: <9208310044.AA10286@relay1.UU.NET> Date: Mon Aug 31 00:14:38 1992 Reply-To: cbmvax!uunet!mullian.ee.mu.oz.au!nsn Sender: Lojban list From: cbmvax!uunet!mullian.ee.mu.oz.au!nsn Subject: A touch of MEX X-To: lojban@cuvmb.cc.columbia.edu X-Cc: nsn@ee.mu.oz.au To: John Cowan Status: RO X-From-Space-Date: Mon Aug 31 00:14:38 1992 X-From-Space-Address: cbmvax!uunet!cuvmb.bitnet!LOJBAN The following is a translation of the first page of _Fundamental Concepts Of Higher Algebra_ by A. Adrian Albert. di'e se fanva fi le pamoi bele'i papri be la'e <> pefi'e la'o <> itu'e CHAPTER I: GROUPS 1. _Sets and mappings._ Abstract algebra is concerned with the study of certain mathematical objects called _algebraic systems_. Each system consists of a set of elements, one or more operations on these elements, and a number of assumptions (about the properties of the elements with respect to the operations) called the _defining postulates_. In this first section we shall introduce some of the elementary notions about sets which form the basis of the precise definitions which we shall present of several algebraic systems. ni'oni'oni'o 1mai cmaci girzu ni'oni'o 1pi'e1mai selcmi ce fancu ni'o le sucta cmacrnalgebra cu srana lenu tadni loi cmaci sibda'i peme'e <> .i ro ciste cu se pagbu loi selci kujo'u su'o se sumti be ri be'o kujo'u loi selru'a {befi lei selkai {belei selci be'o} pelei se sumti be'o} neme'e <> .i vecu'u ledei ckupa'upa'u cu cninyja'o so'o friseljmi sidbo peloi selcmi ge'uku poi jicmu lei satci se xusra co finti be so'o cmacrnalgebra ciste be'o poi se skicu da'e Let \it A be a set whose elements a, b, c, ... are any objects whatever, and let \it B be a second set. Then we say that B is contained in A, and write B {subset} A, if every element of B is in A. If B {subset} A, we call B a subset of A. We may also write A {contains} B and say that A contains B. If A {contains} B and at least one element of A is not in B, we say that B is a proper subset of A and write A {propercontains} B (A properly contains B), or B {propersubset} A (B is properly contained in A). The set having no elements is called the empty set. ni'oca'e ge tauce'afy. ga'e .abu (to fy. sinxa la fraktur. toi) selcmi da nemu'u nau.abu ce by. ce cy. zi'epoi cmima lo'iro dacti gi ce'afy. ga'e by. selcmi gi'enaidu .abu .iseni'ibo go ge by. pagbu .abu gi zo'e ji'u cusku me'o by. na'u klesi .abu gi ro cmima be by. cu cmima .abu .i go meli by. na'u klesi .abu gi by. klesi .abu .idu'ibo ge zo'e ji'u me'o .abu na'u selkle by. gi .abu selpa'u by. .i go ge meli .abu na'u klesi by. gi su'o cmima be .abu na cmima by. gi ge by. nalrolmei klesi .abu gi zo'e ji'u cusku me'o .abu na'u nalrolmemkle by. lo'o.eme'o by. na'u se nalrolmemkle .abu .i le selcmi be noda cu se cmene <> (NOTE: relations aren't part of MEX like operators are: {du}, {klesi} (subset), {cmima} (element of) etc. are bridi, not MEX operators. To incorporate such relations into MEX (for example, when quoting an equation with {me'o}), {na'u} must be used to convert the bridi into an operator. In that case, {me'o re na'u du re} is the equation "2 = 2", and {li re na'u du re} have the evaluated value TRUE.) The intersection of two sets A and B is the set of all elements which are in both A and B. We designate this set by A {intersect} B. If C {subset} A and C {subset} B, then C is called a common subset of A and B. Every common subset of A and B is a subset of A {intersect} B. The union of A and B is the logical sum of A and B. It consists of all elements which are either in A or in B. We designate it by A {union} B. ni'o le selcmipi'i be .abu poi selcmi bei .by poi selcmi cu selcmi ro cmima be .abu .e by .i le go'i cu se sinxa me'o .abu na'u selcmipi'i by .i go meli cy. na'u klesi .abu lo'o.eli cy. na'u klesi by. gi cy. kampu klesi .abu joi by .i ro kampu klesi be .abu joi by. cu klesi li .abu na'u selcmipi'i by. ni'o le selcmisumji be .abu bei by. cu logji sumji .abu by. .i lego'i cu se cmima ro cmima be .abu .a by. gi'e se sinxa me'o .abu na'u selcmisumji by. The concepts of intersection and union may be generalised readily to several sets. Thus if A\1,...,A\n are sets, we define their intersection A\1 {intersect} A\2 {intersect} A\2 {intersect}...{intersect} A\n to be the set of all elements which are simultaneously in every one of the sets A\1,...,A\n. The union A\1 {union} A\2 {union}...{union} A\n consists of all the elements in all the sets A\1,...,A\n. Note the the equation (A\1 {intersect} A\2) {intersect} A\3 = A\1 {intersect} (A\2 {intersect} A\3) = A\1 {intersect} A\2 {intersect} A\3 states that the set consisting of those elements in A\1 {intersect} A\2 which are also in A\3 is the same set as that consisting of those elements of A\1 which are in A\2 {intersect} A\3 and that this set is precisely the set of those elements which are in A\1, in A\2, and in A\3. Similarly, (A\1 {union} A\2) {union} A\3 = A\1 {union} (A\2 {union} A\3) = A\1 {union} A\2 {union} A\3. ni'o lesi'o selcmipi'i je selcmisumji cu frili ke to'erte'i srana za'ure selcmi .i go .abuxi1 celi'o .abuxiny. selcmi gi le sosyselcmipi'i be ro ri be'o no'u li .abuxi1 na'u selcmipi'i .abuxi2 .abuxi3li'o .abuxiny. cuca'e selcmi roda poi cmima role selcmi no'u .abuxi1 celi'o .abuxiny. .i le sosyselcmisumji no'u li .abuxi1 na'u selcmisumji .abuxi2 .abuxi3li'o .abuxiny. cu selcmi roda poi cmima su'ole selcmi no'u .abuxi1 celi'o .abuxiny .i ko jundi lenu go'e .i me'o (vei (vei .abuxi1 na'u selcmipi'i .abuxi2 ve'o) na'u selcmipi'i .abuxi3 ve'o) na'udu (vei .abuxi1 na'u selcmipi'i (vei .abuxi2 na'u selcmipi'i .abuxi3 ve'o) ve'o) .abuxi1 na'u selcmipi'i .abuxi2 .abuxi3 cu xusra lenu le selcmi be ro cmima beli .abuxi1 na'u selcmipi'i .abuxi2 be'o poi cmima .abuxi3 cu du le selcmi be ro cmima be .abuxi1 be'o poi cmima li .abuxi2 na'u selcmipi'i .abuxi3 be'o le selcmi be ro cmima be .abuxi1 .e .abuxi2 .e .abuxi3 .isi'a meli (vei (vei .abuxi1 na'u selcmipi'i .abuxi2 ve'o) na'u selcmipi'i .abuxi3 ve'o) na'udu (vei .abuxi1 na'u selcmipi'i (vei .abuxi2 na'u selcmipi'i .abuxi3 ve'o) ve'o) .abuxi1 na'u selcmipi'i .abuxi2 .abuxi3 --- 'Dera me xhama t"e larm"e, T Nick Nicholas, EE & CS, Melbourne Uni Dera mbas blerimit | nsn@munagin.ee.mu.oz.au (IRC: Nicxjo) Me xhama t"e larm"e! | Milaw ki ellhnika/Esperanto parolata/ Lumtunia nuk ka ngjyra tjera.' | mi ka'e tavla bau la lojban. je'uru'e - Martin Camaj, _Nj"e Shp'i e Vetme_ | *d'oh!*