REVIEW FOR CHAPTER 1

Transcription

1 REVIEW FOR CHAPTER 1 A short summary: I this chapter you helped develop some basic coutig priciples. I particular, the uses of ordered pairs (The Product Priciple), fuctios, ad set partitios (The Sum Priciple) were developed ad used, ad the sigi cat role of fuctios i coutig was itroduced. terms: fuctio; oe-to-oe fuctio; oto fuctio; bijectio; disjoit sets; partitio of a set. Basic Coutig Priciples: The Product Priciple If a ite set S is partitioed ito m blocks of the same size, the S has size m. The Sum Priciple For ay partitio of a ite set S, the size of S is the sum of the sizes of the blocks of the partitio. The Bijectio Priciple Two sets have the same size if ad oly if there is a bijectio betwee them. The Pigeohole Priciple If a set with more tha elemets is partitioed ito blocks, the at least oe block has more tha oe elemet. 1. Ay set with elemets has 2 subsets.

2 REVIEW FOR CHAPTER 2 A short summary: I this chapter, you reviewed the Priciple of Mathematical Iductio i the cotext of uderstadig the uderlyig iductive processes. At rst we used the so-called simple priciple, but this was superseded by the strog priciple which is just as easy to use ad is ofte more coveiet to apply. Oe of the importat uses of mathematical iductio i coutig problems is that it allows use to prove the Geeral Product Priciple. Sice the Product Priciple from Chapter 1 is a special case of the geeral priciple, the term Product Priciple will be reserved for this more geeral oe. terms: iductive process; rst-order recurrece; permutatio; k-permutatio. Basic Priciples: The Priciple of Mathematical Iductio To prove a sequece of statemets idexed by itegers k b it is su ciet to 1. State the sequece of statemets to be proved; 2. (Base Step) Prove the statemet is true for k = b; 3. (Iductive Step) Prove that (for ay N b) the truth of the statemets for k = b, k = b + 1,..., k = N implies the statemet with k = N + 1 is true; 4. (Iductive Coclusio) Coclude by the Priciple of Mathematical Iductio that every statemet i the sequece is true. The Product Priciple Suppose you make a sequece of m choices, where the rst choice ca be made i k 1 ways, ad for each way of makig the rst i 1 choices, the i-th choice ca be made i k i ways, the the total umber of di eret ways to make this sequece of m choices is k 1 k 2 k. 1. The umber of bijectios o [] is!. (Such bijectios are also called permutatios of the set []:) 2. The umber of fuctios from [m] to [] is m. 3. The umber of k-elemet permutatios of [] is ( 1) ( k +1), which ca be deoted as k :

3 REVIEW FOR CHAPTER 3 A short summary: I this chapter you leared how the partitio iduced by a equivalece relatio ca be used i coutig problems. I order to use this techique, you eeded practice i idetifyig the uderlyig equivalece relatio. By the ed of the chapter you had proved the formula for calculatig the umber of k-subsets of [], which is also kow as the biomial coe ciet k, ad the formula for coutig multisets. terms: re exive relatio; symmetric relatio; trasitive relatio; equivalece relatio; equivalece class; biomial coe ciet; ordered-fuctios; multisets. Basic Priciples: Equivalece classes Whe R is a equivalece relatio o the set S, its set of equivalece classes forms a partitio of S. 1. For ay k; 0 with k, the umber of k-elemet subsets of [] is! = k k! ( k)! : 2. For ay k; 0, there are (k + 1) k ordered-fuctios from [k] to []. 3. For ay k; 0, there are +k 1 k ways to choose a k-elemet multiset from []:

4 REVIEW FOR CHAPTER 4 A short summary: I this chapter you were itroduced to the fudametals of graph theory. Trees are amog the most useful types of graphs, ad miimal spaig trees i coected weighted graphs are used i may applicatios. The Bijectio Priciple ad the Priciple of Mathematical Iductio were crucial i the veri catio of importat properties of graphs. terms: graph; degree of a vertex; coected graph; cycle; tree; spaig tree; weighted edge; miimal spaig tree; Prüfer code. Basic algorithm: Dijkstra s Algorithm 1. The sum of the degrees of the vertices i a graph equals twice the umber of edges. 2. Ay tree with vertices has 1 edges. 3. The umber of labelled trees o vertices is 2 :

5 REVIEW FOR CHAPTER 5 A short summary: Picture eumerators were used to solve coutig problems related to multisets ad to motivate geeratig fuctios, which are formal power series whose coe ciets record the elemets of a sequece of umbers. Formal power series ca be used to solve a variety of coutig problems ad to obtai a explicit formula for certai recurreces, ad you saw a example of this techique whe you foud Biet s Formula. I order to work with geeratig fuctios you must uderstad the algebra of formula power series. terms: geeratig polyomial; geeratig fuctio. Basic Priciples: Additio of Formal Power Series! 0 1 a i x i b j x j A = j=0 (a k + b k )x k : k=0 Multiplicatio of Formal Power Series! 0 1 a i x b j x j A = j=0 k=0! kx a i b k i x k : Product Priciple of Picture Eumerators See Problem 204. Product Priciple for Geeratig Fuctios See page (1 x) = + i 1 x i : i 2. Biet s Formula: If F is the -th Fiboacci umber the F = p p! p p! +1 5 :

6 REVIEW FOR CHAPTER 6 A short summary: I this chapter you reviewed the termiology of sets i order to write ad uderstad the Priciple of Iclusio ad Exclusio. You used this priciple to determie the umber of deragemets ad the umber of oto fuctios. terms: set termiology (uio, itersectio, complemet); deragemet. Basic Priciples: The Priciple S of Iclusio ad Exclusio The umber of elemets i i=1 A i is [ A i = X \ ( 1) jsj 1 A i : i=1 S[] ; S6=; The umber of elemets i the complemet of S i=1 A i i a (uiversal) set A is [ X \ A i = jaj ( 1) jsj 1 A i : i=1 S[] ; S6=; i2s i2s 1. The umber of deragemets of [] is X s=0 ( 1) s! s! : 2. The umber of oto fuctios from [k] to [] is X ( 1) s ( s) k : s s=0

7 REVIEW FOR CHAPTER 7 k objects ad coditios recipiets ad mathematical model for distributio o how they are received Distict Idetical Distict k NOT DONE o coditios fuctios (set partitios ito parts) Distict k 1 if k ; 0 otherwise Each gets at most oe k-elemet permutatios P Distict s=0 ( 1)s s ( s) k NOT DONE Each gets at least oe oto fuctios (set partitios ito parts) Distict k! =! 1 if k = ; 0 otherwise Each gets exactly oe permutatios Distict, order matters (k + 1) k P i=1 L(k; i) ordered-fuctios broke permutatios ( parts) Distict, order matters (k) (k 1) k L(k; ) = k (k 1) k Each gets at least oe ordered-oto-fuctios broke permutatios ( parts) +k 1 Idetical k NOT DONE o coditios multisets (umber of partitios ito parts) Idetical Each gets at most oe k subsets 1 if k ; 0 otherwise k 1 1 Idetical NOT DONE Each gets at least oe compositios ( parts) (umber of partitios ito parts) Idetical 1 if k = ; 0 otherwise 1 if k = ; 0 otherwise Each gets exactly oe