ii. O = {x x = 2k + 1 for some integer k} (This set could have been listed O = { -3, -1, 1, 3, 5 }.)

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1 Sets 1 Math 3312 Set Theory Sprig 2008 Itroductio Set theory is a brach of mathematics that deals with the properties of welldefied collectios of objects, which may or may ot be of a mathematical ature, such as umbers or fuctios. The theory is less valuable i direct applicatio to ordiary experiece tha as a basis for precise ad adaptable termiology for the defiitio of complex ad sophisticated mathematical cocepts. Betwee the years 1874 ad 1897, the Germa mathematicia ad logicia Georg Cator created a theory of abstract sets of etities ad made it ito a mathematical disciplie. This theory grew out of his ivestigatios of some cocrete problems regardig certai types of ifiite sets of real umbers. A set, wrote Cator, is a collectio of defiite, distiguishable objects of perceptio or thought coceived as a whole. The objects are called elemets or members of the set. The theory had the revolutioary aspect of treatig ifiite sets as mathematical objects that are o a equal footig with those that ca be costructed i a fiite umber of steps. Sice atiquity, a majority of mathematicias had carefully avoided the itroductio ito their argumets of the actual ifiite (i.e., of sets cotaiig a ifiity of objects coceived as existig simultaeously, at least i thought). Sice this attitude persisted util almost the ed of the 19th cetury, Cator's work was the subject of much criticism to the effect that it dealt with fictios ideed, that it ecroached o the domai of philosophers ad violated the priciples of religio. Oce applicatios to aalysis bega to be foud, however, attitudes bega to chage, ad by the 1890s Cator's ideas ad results were gaiig acceptace. By 1900, set theory was recogized as a distict brach of mathematics. At just that time, however, several cotradictios i so-called aive set theory were discovered. I order to elimiate such problems, a axiomatic basis was developed for the theory of sets aalogous to that developed for elemetary geometry. The degree of success that has bee achieved i this developmet, as well as the preset stature of set theory, has bee well expressed i the Nicolas Bourbaki, Elemets de mathematique (begu 1939; Elemets of Mathematics ): Nowadays it is kow to be possible, logically speakig, to derive practically the whole of kow mathematics from a sigle source, The Theory of Sets. Source: Ecyclopædia Britaica Olie.

2 Sets 2 Naïve Set Theory 1.1 Set Notatio ad Some Defiitios Terms: A set (at this poit i our course) will be thought of as a collectio of objects whose elemets are distiguishable. The objects i a set are called the elemets, or members, of the set. A set is said to cotai its elemets. Covetios/Notatio/Defiitios: We try to use lower case letters for elemets ad upper case letters to deote sets, e.g. A, B, C, X, is read as "is a elemet of". Example: Let A = {1, 2, 5}. 1 A, 2 A, but 3 A. If a set is fiite or has a patter the the set ca be described by listig the elemets. A more geeral way to describe a set is by the use of set builder otatio. Let P(x) be a formula about x or a property of x. The we use { x P(x) } we deote the collectio of elemets x that satisfy P(x). Examples: i. R = {x x is a real umber } This is read as " the set of all x such that x is a real umber " ii. O = {x x = 2k + 1 for some iteger k} (This set could have bee listed O = { -3, -1, 1, 3, 5 }.) Two sets A ad B are equal deoted A = B if ad oly if they have the same elemets. Note: Oe example of why there is issue with the vague defiitio of a set. Cosider R = {X X X }. What is the truth value of R R ad R R?

3 Sets 3 Now more defiitios ad otatio. The uiversal set, which we will deote as U, is the set of all objects uder cosideratio i a give problem sometimes referred to as a super-set. A set A is said to be a subset of set B if ad oly if every elemet of A is also a elemet of B. We use the otatio A B to write that A is a subset of B (ote some books use A B. To idicate that A is ot a subset of B we may write A B. Theorem 1.1 (Properties of ) 1. A A (reflexive property) 2. if A B ad B C, the A C (trasitive property) 3. A B ad B A if ad oly if A = B (atisymmetric property) represets the empty set or ull set, which is defied as the set with o elemets. Theorem 1.2 The empty set is a subset of ay set. Corollary 1.3. A oempty set has at least two subsets. Corollary 1.4. There is at most oe empty set.

4 Sets 4 Example: The umber of 2 elemet subsets of a set with elemets is ( 1) deoted by = 2 2 Let us examie this by example first. Let A = {a, b, c}. The collectio of all 3 3(3 1) 2-elemet sets is { {a,b}, {a,c}, {b,c} }, ad = = Theorem 1.5: The umber of k-elemet subsets of a set with elemets is! deoted by = k k!( k)! Theorem 1.5: If is a oegative iteger ad k, = k k Give a set S, the power set of S is the set of all subsets of the set S. The power set of S is deoted by P(S). Examples: 1. Let A = {a, b, c}. Describe the set P(A). Solutio: P (A) = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }. 2. P ( ) =?

5 3. Let A be ay set. True/False. Sets 5 1. A 2. { } A 3. P(A) 4. { } P( A) 5. A Theorem: The umber of all subsets of a set with elemets is = 2 k = 0 k Let S be a set. If there are exactly distict elemets i S, where is a oegative iteger, we say S is a fiite set ad is the cardiality of S. The cardiality of S is deoted by S. A set is said to be ifiite if it is ot fiite. I our ew otatio Theorem: If A =, the P(A) = k = 0 = k 2 Examples 1. Let A = {1, 2, 5}. The A = Let S be the set of letters i the Eglish alphabet. The S = = 6. { } = 4. {1000} = 7. { } = 5. {1,2,,1000} = 8. {1,2,1000} = Later i the course we will discuss cardiality more formally ad for ifiite sets.

6 Homework: 1. Prove the trasitive property of. (Part 2 of Theorem 1.1) 2. Explai why { } 3. Give a example of a set X such that X has a elemet that is also a subset of X. 4. List the elemets of P ( ) 5. How may elemets does P (P ( ) ) have? list the elemets. 6. How may elemets does P (P (P (P ( ) ))) have? 7. How may 2-elemets subsets does P (P (P (P ( ) ))) have? 8. Prove {{x}, {x,y}} = {{z}, {z, t}} if ad oly if x = z ad y = t. Sets 6