7. Counting Measure. Definitions and Basic Properties

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1 Virtual Laboratories > 0. Foudatios > Coutig Measure Defiitios ad Basic Properties Suppose that S is a fiite set. If A S the the cardiality of A is the umber of elemets i A, ad is deoted #( A). The fuctio # is called coutig measure. Coutig measure plays a fudametal role i discrete probability structures, ad particularly those that ivolve samplig from a fiite set. The set S is typically very large, hece efficiet coutig methods are essetial. The first combiatorial problem is attributed to the Greek mathematicia Xeocrates. I may cases, a set of objects ca be couted by establishig a oe-to-oe correspodece betwee the give set ad some other set. Naturally, the two sets have the same umber of elemets, but for some reaso, the secod set may be easier to cout. The Additio Rule The additio rule of combiatorics is simply the additivity axiom of coutig measure. If {A 1, A 2,..., A } is a collectio of disjoit subsets of S the #( i =1 A i ) = #( A i =1 i ) Simple Properties The coutig rules i this subsectio are simple cosequeces of the additio rule. 1. Show that #( A c ) = #(S) #( A). Hit: A ad A c are disjoit ad their uio is S.

2 2. Show that #( B A) = #( B) #( B A). Hit: A B ad B A are disjoit ad their uio is B. 3. Show that if A B the #( B A) = #( B) #( A). Hit: Apply the differece rule ad ote that A B = A 4. Show that if A B the #( A) #( B). Thus, # is a icreasig fuctio, relative to the subset partial order o P(S), ad the ordiary order o R. Iequalities This subsectio gives two iequalities that are useful for obtaiig bouds o the umber of elemets i a set. 5. Suppose that {A 1, A 2,..., A } is a a fiite collectio of subsets of S. Prove Boole's iequality (amed after George Boole): #( i =1 A i ) #( A i =1 i ) d. Defie B 1 = A 1 ad B i = A i ( A 1 A i 1 ) for i {2, 3,..., }. Show that {B 1, B 2,..., B } are pairwise disjoit ad have the same uio as {A 1, A 2,..., A }. Use the additio rule Use the icreasig property. Ituitively, Boole's iequality holds because parts of the uio have bee couted more tha oce i the expressio o the right. 6. Suppose that {A 1, A 2,..., A } is a fiite collectio of subsets of S. Prove Boferroi's iequality (amed after Carlo Boferroi): #( i =1 A i ) #(S) (#(S) #( A i =1 i )) Hit: Apply Boole's iequality to {A 1 c, A 2 c,..., A c } ad use DeMorga's law. The Iclusio-Exclusio Formula 7. Show that #( A B) = #( A) + #( B) #( A B) Note first that A B = A ( B A) ad the latter two sets are disjoit. Now use the additio rule ad the differece rule.

3 8. Show that #( A B C) = #( A) + #( B) + #(C) #( A B) #( A C) #( B C) + #( A B C) Hit: Use the iclusio-exclusio rule for two sets. You will use this rule three times. Exercise 7 ad Exercise 8 ca be geeralized to a uio of sets; the geeralizatio is kow as the iclusio-exclusio formul 9. Suppose that {A i : i I} is a collectio of subsets of S where #( I) =. Show that #( A i I i ) = ( 1) k 1 # k =1 ( J I ) ad (#( J) =k) ( A j J j ) The geeral Boferroi iequalities, amed agai for Carlo Boferroi, state that if sum o the right is trucated after k terms (k < ), the the trucated sum is a upper boud for the cardiality of the uio if k is odd (so that the last term has a positive sig) ad is a lower boud for the cardiality of the uio if k is eve (so that the last terms has a egative sig). The Multiplicatio Rule The multiplicatio rule of combiatorics is based o the formulatio of a procedure (or algorithm) that geerates the objects to be couted. Specifically, suppose that the procedure cosists of k steps, performed sequetially, ad that for each j, step j ca be performed i j ways regardless of the choices made o the previous steps. The the umber of ways to perform the etire algorithm (ad hece the umber of objects) is 1 2 k. The key to a successful applicatio of the multiplicatio rule to a coutig problem is the clear formulatio of a algorithm that geerates the objects beig couted, so that each object is geerated oce ad oly oce. That is, we must either over-cout or uder-cout. The first two exercises below give equivalet formulatios of the multiplicatio priciple. 10. Suppose that S is a set of sequeces of legth k, ad that we deote a geeric elemet of S by ( x 1, x 2,..., x k ). Suppose that for each j, x j has j differet values, regardless of the values of the previous coordiates. Show that #(S) = 1 2 k 11. Suppose that T is a ordered tree with depth k ad that each vertex at level i 1 has i childre for

4 i {1, 2,..., k}. Show that the umber of vertices at level k is 1 2 k. Product Sets 12. Suppose that S i is a set with i elemets for i {1, 2,..., k}. Show that #(S 1 S 2 S k ) = 1 2 k 13. Show that if S is a set with elemets, the S k has k elemets. 14. Show that the umber of ordered samples of size k that ca be chose with replacemet from a populatio of objects is k. Fuctios ad Bit Strigs 15. Show that the umber of fuctios from a set A of elemets ito a set B of m elemets is m. Hit: To costruct f : A B, there are m choices for f ( x) for each x A. Recall that the set of fuctios from a set A ito a set B (regardless of whether the sets are fiite or ifiite) is deoted B A. The result i the previous exercise is motivatio for this otatio. 16. Elemets of {0, 1} are sometimes called bit strigs of legth. Show that the umber of such strigs is 2. Subsets 17. Suppose that S is a set with elemets. Show that there are 2 subsets of S. Hit: To costruct A S, decide whether x A or x A for each x S. 18. Suppose that {A 1, A 2,..., A k } is a collectio of subsets of a set S. Show that there are 2 2k differet (i geeral) sets that ca be costructed from the k give sets, usig the operatios of uio, itersectio, ad complemet. These sets form the algebra geerated by the give sets. Show that there are 2 k pairwise disjoit sets of the form B 1 B 2 B k where B i = A i or B i = A i c for each i. Argue that every set that ca be costructed from {A 1, A 2,..., A k } is a uio of some (perhaps all, perhaps oe) of the sets i (a). 19. I the Ve diagram applet, observe the diagram of each of the 16 sets that ca be costructed from A ad B. 20. Suppose that S is a set with elemets ad that A is a subset of S with k elemets. Show that the umber of subsets of S that cotai A is 2 k. Computatioal Exercises

5 21. A licese umber cosists of two letters (uppercase) followed by five digits. How may differet licese umbers are there? 22. Suppose that a Persoal Idetificatio Number (PIN) is a four-symbol code word i which each etry is either a letter (uppercase) or a digit. How may PINs are there? 23. I the board game Clue, Mr. Boddy has bee murdered. There are 6 suspects, 6 possible weapos, ad 9 possible rooms for the murder. The game icludes a card for each suspect, each weapo, ad each room. How may cards are there? The outcome of the game is a sequece cosistig of a suspect, a weapo, ad a room (for example, Coloel Mustard with the kife i the billiard room). How may outcomes are there? Oce the three cards that costitute the outcome have bee radomly chose, the remaiig cards are dealt to the players. Suppose that you are dealt 5 cards. I tryig to guess the outcome, what had of cards would be best? 24. A experimet cosists of rollig a fair die, drawig a card from a stadard deck, ad tossig a fair coi. How may outcomes are there? 25. A fair die is rolled 5 times ad the sequece of scores recorded. How may outcomes are there? 26. Suppose that 10 persos are chose ad their birthdays recorded. How may possible outcomes are there? 27. I the card game Set, each card has 4 properties: umber (oe, two, or three), shape (diamod, oval, or squiggle), color (red, blue, or gree), ad shadig (solid, ope, or stripped). The deck has oe card of each (umber, shape, color, shadig) cofiguratio. A set i the game is defied as a set of three cards which, for each property, the cards are either all the same or all differet. How may cards are i a deck? How may sets are there? 28. A fair coi is tossed 10 times ad the sequece of scores recorded. How may sequeces are there? How may sequeces are there that cotai exactly 3 heads? 29. A strig of lights has 20 bulbs, each of which may be good or defective. How may cofiguratios are there?

6 30. The die-coi experimet cosists of rollig a die ad the tossig a coi the umber of times show o the die. The sequece of coi results is recorded. How may outcomes are there? How may outcomes are there with exactly 2 heads? 31. Ru the die-coi experimet util exactly 2 heads occurs. 32. Suppose that a sadwich at a restaurat cosists of bread, meat, cheese, ad various toppigs. There are 4 differet types of bread (select oe), 3 differet types of meat (select oe), 5 differet types of cheese (select oe), ad 10 differet toppigs (select ay). How may sadwiches are there? The Galto Board The Galto Board, amed after Fracis Galto, is a triagular array of pegs. Galto, apparetly too modest to ame the device after himself, called it a quicux from the Lati word for five twelfths (go figure). The rows are umbered, from the top dow, by (0, 1, 2,...). Row k has k + 1 pegs that are labeled, from left to right by (0, 1,..., k). Thus, a peg ca be uiquely idetified by a ordered pair (i, j) where i is the row umber ad j is the peg umber i that row. A ball is dropped oto the top peg (0, 0) of the Galto board. I geeral, whe the ball hits peg (i, j), it either bouces to the left to peg (i + 1, j) or to the right to peg (i + 1, j + 1) The sequece of pegs that the ball hits is a path i the Galto board. 33. Show that there is a oe-to-oe correspodece betwee each pair of the followig three collectios: Bit strigs of legth Paths i the Galto board from (0, 0) to a peg i row. Subsets of a set with elemets. Thus, from the previous exercise, each of these collectios has 2 elemets. 34. Ope the Galto board applet. d. Move the ball from (0, 0) to (10, 6) alog a path of your choice. Note the correspodig bit strig ad subset. Geerate the bit strig Note the correspodig subset ad path. Geerate the subset {2, 4, 5, 9, 12}. Note the correspodig bit strig ad path. Geerate all paths from (0, 0) to (4, 2). How may paths are there? Virtual Laboratories > 0. Foudatios > Cotets Applets Data Sets Biographies Exteral Resources Keywords Feedback