MA1 Chapter 1 PDF

~ MATHEMATICAL ANALYSIS 1 ~ I Chapter  ~ BASIC NOTIONS OF SETS THEORY ~ [ WHAT IS... A SET ? SETS are...

~ MATHEMATICAL ANALYSIS 1 ~ I Chapter  ~ BASIC NOTIONS OF SETS THEORY ~ [ WHAT IS... A SET ? SETS are PRIMITIVE concepts , that is , concepts that we assure to be UNIVERSAL. We can somehow describe what do we mean with the word SET but we do NOT , define SETS as a general concept. 3 The PRIMITIVE CONCEPT of SET is then used to build the WHOLE THEORY of the course !!  N[gyµ!%ÉÉ ~ NOTABLE EXAMPLES ~ I of sets y IN := { 942 } nature numbers }.. ,. ↑ = { 92,3 ,... } Definitions - ~ :-& I { 0 , -1-1,1=2 ,... } integer numbers " such that it ← Q := { q = ÷ : 2- c- I end nt1N+ } rational numbers ↑ " " in ,, , belong to , , 113 real numbers } : next etoisodes :) ① : complex numbers <  ~ OPERATIONS WITH SETS ~ Fix an ambient set ✗ and three SOBSETS A ,B C , ≤ ×. with sets) De_f(Basic operations ✗ Ac ↑ or EA := { ✗ c- ✗ : ✗ ¢A } C ↑ complement " does not belong 6- ¢ An B := { ✗c- ✗ : ✗c- A and ✗ c- B} B ↑ A intersection AUB := { ✗ EX : ✗c- A or ✗ c- B } A onion A - B := { ✗ c- A : ✗ at B } T difference  - EXERCISES ~ 1. List the elements of the sets : Zay -14=0 ] 2 3=8 × } + u2=u3 - IN ④ { ✗ c- : × < 10J , { ✗ c- IR : ✗ , { u c- : 3. List the elements of AUB , An B end A - B , where : A = { ✗C- 2 : ✗ 2 P) v Q is false , while PVQ is true. ↑ " " " 2 is odd or 3 is even " 2 is even or 3 is even  LOGIC CONNECTORS : , " " " equivalence " means implication , means " P " " " Q is read P implies Q or if P then Q , means that if P is true then Q is true as well. P Q is read " P is equivalent to Qi, and means * a) ^ (Q P ) ¥ Convince yourself that P means - Pv Q. € Let a ≥o. Show that : 1×1 ≤ a - a. ≤ × ≤ a 1×1 > a ✗< - ou v × > on  - PREDICATES ~ ✗ Roughly speaking , a PREDICATE on a set is a statement P that associates to each ✗C-✗ exactly eve FORMULA Pcx). In other words , a PREDICATE is a FUNCTION P : × → { formulas } " " " Ey. ✗ = -1N , Pln) = n> 1 ". Then : Plot = o > 1 False " " P (2) = 221 True ✗ = R , f IR : → R. The predicates " " " E. G) = f- (x) = o and I G) = ft) > 0 ,, are called EQUATION and INEQUALITY , respectively.  - EXERCISES ~ I. Solve the following equations in 112 : ✗2-1×-2 = 0 ✗ 3- ✗2- 2X = 0 1×1 = ✗ 2. ! Solve the following inequalities in 112 2+3×+2 3-1×-0 × ≤ 0 ≥ 0 ✗ < 0 ✗ " " " " 3. Let PG) = × > 2 and G) = ✗ 2- × > 0 and ✗ = R. Say if the following formulas are true or false : Pco) Pco ) v C-1) QC2) TPC 1) - v Qcz) PCZ) ② (3) ② (1) pct) Qco) (3) [ QU) =) (2) ] Plo) n QC 1) -  - QUANTIFIERS ~ Many formulas and predicates say that a certain property holds for element at least element of set every one or a. " 2 " EI 1: for every real number there holds ✗ ≥o. 2 : " there exists a reel number such that ✗ 2--4 ". The It denotes the " symbol universal quantifier and means for every y. The the existential " there exists " symbol 7 denotes quantifier and means. So , the predicates I and 2 can be mathematically written as 1 : f- ✗ c- IR X2 ≥ 0. , 2 : 3- ✗ ER : ✗ 2=4.  - QUANTIFIERS ~ More if A is and P generally , a set is a predicate over A , then f- ✗ c- A Pcx) means that PG) is true for ✗c- A , every. 3- ✗ c- A Pc means that PGD is true for at least one ✗ c- A. , RMk_ To show that it ✗ c-A , P G) is false , it is enough to find Pca) false Such element is celled some a c- A : is. a counterexample. For instance : the /N , F KEN : t2= n is false Indeed , there is not KEN such that k 2--2 (next episodes). ↑ n=2 is a counterexample !  - EXERCISES ~ Write in formal the following formulas and is a way say they are true or false. 1. There exists a nature number greater than 3. ↳ In c- IN : n> 3 (True). 2. The then the number itself square of a nature number is greater _ 3. There exists a natural number smaller than its cube. 4. There exists than all the others a naturel number greater. real 5. Every number is the sum of two rational numbers. 6. The only real solution of ✗+3=0 is -3. The rational square of every 7-. rational number is.  - THE NATURAL NUMBERS ~ The set of natural numbers ( denoted with 1N ) is one of the most important sets in the whole Mathematics. It is made of elements ( the natural numbers) that you can USE very well. However , are you able to define IN ? Informally , we usually write IN : = { 0 , I , 2 , 3 ,. - - - }. Even if it is intuitive , this definition hes some problems , e. g. : The list 0,1 , 2,3 ,.. _ is not complete. Implicitly , we are saying that IN is made of the number 0 and any other number obtained by counting forward indefinitely. r " But , what does counting forward indefinitely ,, Meen ? " How do we know that we on keep counting forward indefinitely ? ,,  - THE NATURAL NUMBERS ~ A way of overcoming these kind of problems is following the Peano 's axiomatic approach. Intuitively , we went to build a set which satisfies the following properties : IN is not ! 1 : ° is a natural number ← empty us to allows. This count ! T 2 : the successor of eny naturel number is a naturel number. → 0 is the smallest 3 0 is not the successor of natural number any :. naturel number ! → 4 Different naturel number have different successors No ! :. cycles 5 If subset A- ≤ IN contains 0 and the successor of any : a element a c- A then 1- = IN. " This , is called the Induction Principle "  - THE NATURAL NUMBERS ~ In formal terms , the Peano 's axioms read as follows : Z IN satisfying the AO. a set following properties : A1. 0 C- IN. function) " A- 2. Z S : IN → IN n c- IN ↳ Scu> c- IN ( the successor " : , A 2. t.tn c- IN Scu) =/ 0. , A- 2. 2. V-n.ME IN n≠m sch) =/ San). , A- 3. tA≤ IN , { ( ◦ c- A) a ( the A scaled ) } A- = IN. In this the set of natural numbers be defined as way , can IN : = { 0 , Slo) , SCS lol ) , SCS (SC) )) ,.... } { " = 0 , I , 2 , 3 ,. _. } if we county in bese 10.  ~ EXERCISES ~ I. Using only A1 and A2 , show that 3 C- IN. 2. Consider the set A = { 0,1 , 2,3 } with the assumptions : Slo) =L , s (1) = 2 , S / 2) =3 and s (3) = 0. Show that woe system obeys axioms A1 and A2. 3. Using only Ah , A- 2 end A- 2.1 , show that 41--0. 4. Consider the set A = { 0,1 , 2,3 } with the assumptions : Slo) =L , s (1) = 2 , S / 2) =3 and s (3) =3. Show that A2 end 1-2.1 woe system obeys axioms Ah ,. 5. Using only Ah , A- 2 , A- 2.1 and 1-2.2 , show that 6=12. 6. Consider the set A- = { 0,42 , 1 , % , 2 ,... }. Show that A- 2.1 end A2 2 woe system obeys axioms Ah , A2 , _.  THE INDUCTION PRINCIPLE ~ The test of Peano 's axioms reads as A- 3 : tA≤ IN , { IOEA ) r ( the -1 son)€A ) } A- = IN. In more practical words , 1-3 con be stated as follows : INDUCTION PRINCIPLE ( AT) : Let P be a predicate over IN. If Plot is true end whenever Pcu) is true Pentad is also true , , then Pcu) is true for every nc-IN.EE Let us show that n Cnts) tf n c- IN 01-1-12 +... + n =. , = E- Convince yourself that 1-3 end AT eve equivalent.  ~ EXERCISES ~ I. Show that V- newt , 11-3 +... + 2h - I = n? L. Show that th C- IN , 2-14-1. _. + 2h = n (nts). " 3. Show that the 1N , for≥ -1 , @ +a ) ≥ It now. " 4. Show that if 1^-1 __ n , then IPC A) 1--2. 3. Show that the 1N , 02+12-1... + n2= nCnt1)C2h. 6 6. Define n ! ( n factorial ) as i o ! = I n ! = n.cn 1) ! - , the /Nt. Show that : 6.1. An ≥ 4 , n! > 2h. " 6. 2.tn ≥ 1 , n ≥ n !  ~ EXERCISES ~ 1. Show that A- { 0,42 , 1 , % , 2 ,... } =/ IN. 2. ADDITION in 1N. Let ME IN. We define = } 0 -1M : = m ◦ +m=m , ^ᵗM=s↳)+m:= scmj , ME IN , Scu) + m : = s(n+m ). Itm = SH) -1M = S ( Itm) ,. _. _. ↓ show that ! Informally , Ktm means to " V- MEIN Mto = m. increment M of K units ,. , th , MEIN ntscm) = S ( htm). , Deduce that Scu) = nts and that n+m=m+n ( + is commutative ). them , KEN , n + ( Mtk ) = ( utm) +K (+ is associative]. th , un , k C- IN if ntk = mtk n=m ( cancellation law ). 3. What about MULTIPLICATION in 1N ?? Check T TAO 's > Book / Section 2. 3) !!  ~ BINOMIAL THEOREM ~ As all of you know , for every pair a. b. c- IR , we have : Cat b) 2= ◦ ' ( at b) = 1 , let b) = atb a2+ Zab + b2. , theorem of the The binomial is a generalization formulas above. ( Binomial thin) Let ,b IR new TIM a c- and. Then : n ( at b) " = [ (1) akb " - k k=o where , by definition , ( L) : = knew and KEIN : Ken. ☐ , k ! Cn K) - ! ← Binomial coefficient in [ Ck : = Co + Cat - - - t Cu , 1- Cu Co , _.. , Cu C- IR. , - k= 0 ← Son of a sequence  ~ THE INTEGER NUMBERS ~ We to have seen how axiomatically construct the set IN and how to define the sum between natural numbers. However ) " , notice that the difference ( the inverse ,, of the sum is not well - defined in 1N. For instance , 2- 3 = - I ¢ IN ?! For this reason , we introduce ( informally ) the set of the integers : I := {... , -2 , -1,0 , 1 , 2 ,.. _ } Roughly speaking " , I is obtained by 1N , by adding ell the opposites " - h of every n c- IN , with the convention - o :=0. could extend IN to build Rinks One proceed formally and properly I. Check T Tao 's - book ( Chp 6J ! Also the multiplication is well - defined in I : it works as in IN " with the product of the signs ,, convention.  ~ THE RATIONAL NUMBERS ~ We difference operation 1-) " have seen that IN is not closed ,, under the , while I is ( this is why we have introduced the set 2). " Similar I is not closed , under the division operation ( :). This is why we ( informally) introduce the set of rational numbers ☒ := { 9=-4 : n , me 2 with 2- to ]. Roughly speaking , we consider the set of all fractions ( or quotients) which " obtain one can by dividing , two integers. The 4 basic operations are defined as follows : É 9- ± g- ÷ bd , % g-. : = I td , % :& : = ÷ { - RIK By definition , 9- ~ ÷ ( 9- and g- are equivalent ) rift ad = ab. For instance , ÷ , ~ ¥ ( in general , we just write ÷ %) =.  ~ EXERCISES ~ I. Show that if % ~ ÷ and § ~ ¥ , then : % }. ~ g- ¥ - ; 9-+5 - E- + ¥. that is , a- b = ¥ , with ✗ c- IN c- Nt , y 2. Deff ( Ordering ) Let a.b c- ☒. We say that I a> b riff a- b is a positive rational ; a < b ist b- a is a positive rationale ; @ ≤ b riff either acts or a -_ b. Show that for xcy 2- c- ☒ have every we = , , Exactly one of ✗ =y , ✗< y or ×> y is true ; ✗ < y ist y> × ; If ✗ Cy end y < 2- , then ✗ < 2- ; If xcy and then ✗ 2- < 2-so y 2- ,. ~ NOTABLE EXAMPLES ~ 1 IN := { 0,112 ,... } natural numbers I := { 0,1=1,1--2 ,... } integer numbers Q :={q=÷ : 2- c- I end nt1N+ } rational numbers - % " r.lt#ttED → ° I & ? IE ,  The set ④ of rational number FACT : all is - beautiful and !! disappointing at the same time Why beautiful : ① It is countable , that is , I ☒I = / IN / (next episode) ② IQ is closed wider the 4 basic operation (+ , - , , :) Why disappointing : " ⑦ Easy , eqn 's like ✗ 2- 2 = 0 are not solvable in ☒. # ② It is not possible to construct a one-to-one correspondence between the set ⊕ and the line In other words µ. , 6 In mathematical words is NOT COMPLETE !! ,  Problem ② is every series issue in Measure Theory : We expect that the Ieught → of ANY segment can be obtained ) as a multiple of a UNIT leaflet segment (no Walter how we choose the unit ⇐ d ' " 2 e d = one (E) = s between the ÷ ' " " " " "" The ratio and the edge diagonal of a square is T2 ¢ TttM(Property d. 3) If p ≥o satisfies p2=2 , then p ¢ ☒. Ff By : contradiction ,...  - THE SET of REAL NUMBERS 112 ~ Roughly speaking , the set of red numbers is obtained from ⊕ irrational numbers that ell niubers by adding all , is having non - periodic infinite decimal expansion ( V2 , B , it , e , -.. ). MAIN PROPERTIES : The 4 operations (+ :) and the order relation < defined - , , , on ②~ extend to Pr. Actually , new operations like T are well - defined in IR. the set R is complete : geometrically , this means that there is a one-to-one correspondence between 112 aid a line. ~ > This property will allow to do calculus stuff , like limits and derivatives. build R the set ⊕ Gsimi her RMk_ It is possible to by extending to to 2 from I to Q ) whet pass from we did To IN and. , However , we will follow an axiomatic approach.  - THE SET of REAL NUMBERS 112 ~ As said , the set 112 will be constructed axiomatically. This means that we will assume the existence of a set ( that we will cell D) satisfying the following properties ( axioms) : AO There exists set 112 : a satisfying the following properties : A1 : The operation + ( the sum) is well - defined end selisf.es : + is commutative % : ta b c- IR , atb = b+a ; ← , + is associative ← % : tactic c- R , ( at b) + c = a+ ( b + c) , 0 is the neutral Sz 0€ IR end for c- IR a -10 a ← , : = element for , the sum Sa tack a c- IR end at C- a) ◦ " opposite ota - = : ,. E- a is the " RMI : she allows us to define the difference in R , by Taib c- 112 , a - b := a -1 C- b).  - THE SET of REAL NUMBERS 112 ~ A2 : The operation. ( the product ) is well - defined end selisfies : is commutative Pr : ta b. c- IR , a. b = b. w ; ← , is associative ← Pz : ta b. c , c- R , ( a. b) - c = a. ( b. c) ; 1 is the neutral Pz I EIR end fat IR a. 1 a ← , : = element for , the product ^ Pu : Fat R - { } o a- 1ER and a. a- = 1. , RMI : Pie allows us to define the division in R by , tack V-b-to.ee/b:--- a - b- 2. = a-. , b  - THE SET of REAL NUMBERS 112 ~ 1-3 : An order relation ≤ is well-defined and satisfies : " 01 V-ac-IR.ae a ≤ is reflexive " antisymmetric : ; ← ≤ is d 02 : tab c- R , if a ≤ b end b≤ a a = b ; ≤ is transitive ← 03 : ta ,b , c c- R , if a≤ b end b≤ C as c ; " total 04 : Taib c- IR , either a ≤ b or b. ≤ a. ← ordering , A- 4 : t , and ≤ satisfy : SP : Taib , c c- IR a. ( btc ) = ee - b + a. c ; , so : tails c c- IR if a ≤ b at C ≤ b. to , , ; Po : ta ,b c- ID if a -30 end b ≥o ab ≥ 0. ,  - EXERCISES ~ I. Show that =. the R a. 0 = 0 and - a = C- 1). er. ,. tack , tb -1-0 , _ a- b = ÷ = % that IR : a ≥o - a ≤ 0. Ya , b c- IR , 7s ≥o as ≤ bs. 2. Show that the set ☒ satisfies to - A ↳. RMI From Ex 2 we deduce that the axioms AO - AG are NOT snltrcieut to define the set 112 = this is because ☒ obeys all of them but , as we have seen , V2 # ☒ ! ! ns To introduce the last axiom of heel numbers , we here to give the notion of intervals.  ~ INTERVALS ~ Fix a.b c- 112 note that a ≤to. We define = closed } I [ rib ] : = { ✗c- 112 : a ≤✗ ≤ b } interval BOONDED Pen { } ( ab ) : = ✗ c- Pr : a< ✗ < b - interval closed } [b.too) : = { ✗ c-R = ×≥ by ← interval DED ON BOON too a), : = { ✗ c- 112 : ✗< a } - open interval Thinks : ⑨ We may also write Ja ,b[ , Taib] ,... in piece of (aib) (a) b] , ,.... ② to (+ infinity) and - A C- infinity) are symbols , not numbers. = in _ " denotes Smith greater " denotes smile krone than " real ruber any real ruber , negative then any ,  Fis 32,1-00 [ n [-3,5 ] = { * c- IR :.. _. } IR - [-1,1) = { ✗ c- 112 : - - - } 113+0112 _ = / ✗ c- R :... I [Otro ) - IN = { ✗ c- R : - - - } [ -7 3) , U ( 7,13 ] n 1-1,9 ) = { ✗ c- R : - - - / (1-2,1) 0 ( 6,9 ) )c= { ✗ c- R : - - - - } A B Compute An B , AUB and - for : A- = 1- no , 2) 13=(1,1-0). , 1- = f- 110 ] , B = 10,1 ] - 3. ✗ ER ✗2