Teacher s Marking. Guide/Answers

Transcription

1 WOLLONGONG COLLEGE AUSRALIA eacher s Markig A College of the Uiversity of Wollogog Guide/Aswers Diploa i Iforatio echology Fial Exaiatio Autu 008 WUC Discrete Matheatics his exa represets 60% of the total subject arks Readig ie: 5 iutes ie allowed: hours DIRECIONS O CANDIDAES Refer to age IC Educatio Ltd tradig as Wollogog College Australia CRICOS 07D ABN 4059 WCA-WUC-EXSF age of

2 SAR OF EXAM uestio (0 arks) Marks (a) (i) Stateet rue (ii) Not a stateet sice it is a questio (iii) Stateet False If x = 0 xy = xz does ot iply y = z (b) eacher s Markig F F F Guide/Aswers (i) R (( ) R) ( ) F F F F F F F F F F F F F F F F F F F F F F F F F F F F ark for each colu correspodi g to each of the pricipal coectives suitable adjustets for errors = 4 (ii) A cotiget stateet sice it ca be either true or false depedig o the truth values of the costituet parts (c) (i) ( ) ( ) F F F Sice caot be both true ad false there is o cobiatio of ad F with F o the ai coective herefore the stateet is a tautology for cocludig stateet WCA-WUC-EXSF age of

3 WCA-WUC-EXSF age of eacher s Markig Guide/Aswers (ii) (Associative Law) (Coutative Law) (Distributive Law) (Double Negatio Law) Law) De Morga's ( (Double Negatio Law) Law) (De Morga's (Iplicatio Law) herefore the stateet is a tautology [Naes of laws ot required for full arks] (d) (i) here is o largest atural uber or for every atural uber there is aother atural uber which is oe bigger Every atural uber is oe less tha a give atural uber (ii) is true ad is false (e) (i) + < (ii) ( + < > < < uestio (0 arks) (a) oes) (Modus Syllogis) (Law of R R R R (b) (i) Couterexaple: = is prie ad is ot odd herefore the stateet is false

4 (ii) roof: ( x < 4 x > 4) x > 4 x > 6 x 5 > (iii) his stateet is true For exaple: x = = > x x x > (iv) False eacher s Markig Couterexaple: x = y = = but x y (c) "x ( Guide/Aswers ) x 0 x x = 0 x + x (d) o prove: " Ù + + is odd For eve let = k k + + = ( k) = 4k = + k + + k + ( k + k) + = l + l herefore for eve + + is odd For odd let = = ( + ) = 4 = 4 = ( + + ) = p + p + (e) (f) herefore for odd + + is odd herefore " Ù + + is odd { } = E = { x N : < < } A = x U = = { 0 } A B = { 0} ad = { x : x < } (i) 0 C False 0 > 0 C (ii) Ø C rue (iii) B A = {} 0 rue C WCA-WUC-EXSF age 4 of

5 (iv) { } ( A) False { } ( A) ad { } ( A) (v) B = { x : x < 0} { x : x < 0} or 0 C B but 0 { x : x < 0} C False Couterexaple: C B but uestio (0 arks) (a) (i) Let A ad B be sets he A ad B are said to be disjoit if A B = Ø (ii) Let A ad B be subsets of a uiverse U he the differece of A ad A B = x U : x A x B is give by { } B eacher s Markig (b) Let x A B x A B x A x B ( Defiitio of set differece) ) Guide/Aswers B x A x B x A B x A ( Defiitio of itersectio) A B A B Let x A B x A B x A B x A x B x A x B x A B ( Defiitio of set itersectio) ( Defiitio of set differece) A B A B A B = A B [Naes of laws ot required for full arks] (c) o prove A ( A B) = A B A ( A B) A ( A B) ( Defiitio of set differece) A ( A B ) ( De Morga's Law) ( A A) ( A B ) ( Distributive Law) φ ( A B ) ( Iverse Law) A B ( Idetity Law) ( Defiitio of set differece) = A B [Naes of laws ot required for full arks] (d) A = { 4 5} ad = { 4 5 6} = {( a b) A B : a < b} B R WCA-WUC-EXSF age 5 of

6 (i) A B = {( 4) ( 5) ( 6) ( 4 4) ( 4 5) ( 4 6) ( 5 4) ( 5 5)( 5 6) } (ii) {( 4)( 5)( 6)( 4 5) ( 4 6) ( 5 6) } R = (iii) Sketch of graph of R: 6 x x x 5 x x 4 x eacher s Markig Guide/Aswers (iv) R = {( 4 ) ( 5 ) ( 5 4) ( 6 ) ( 6 4) ( 6 5) } (v) No For first eleets ad 4 there is ore tha oe secod eleet (e) F = {( y) : y = x } x (i) he doai of F is all of ad for each eleet x there is oly oe value of y x x x y = x y x = x his is evidet fro the horizotal lie test (ii) F is oe-to-oe: ( ) ( ) F is oto: Rage of F is Sice F is oe-to-oe ad oto F = = uestio 4 (0 arks) {( x y) : x = y } {( x y) : y = ( x + ) } (a) S = {( x y) : xy 0} F is a fuctio WCA-WUC-EXSF age 6 of

7 x x x = x 0 herefore S is reflexive x y xy 0 yx 0 herefore S is syetric x y z xy 0 yz 0 does ot iply xz 0 Couterexaple x = y = 0 z = he xy = 0 yz = 0 xz = < 0 herefore S is ot trasitive (b) = { 0 4 5} A (i) = ( 0 )( 5) eacher s Markig (c) (ii) = (iii) ( 4 ) = 4 Guide/Aswers (i) Associativity uder : "a b c Ù c a b c = a b (ii) Law of richotoy: If a b Ù the oe ad oly oe of the followig relatioships hold: a < b a = b a > b (iii) Well Orderig riciple: If A is a o-epty subset of Ù the A has a least eleet (d) 0 is ot wellordered (e) [ 0 ) = { x : 0 x < } is ot well-ordered he o-epty subset ( 0 ) has o least eleet herefore [ ) a b b = al b a = a l = a b l (f) CLAIM (): ( ) CLAIM (): = herefore CLAIM () is true Assue CLAIM (k) is true k ie ( ) k ie k = l l () WCA-WUC-EXSF age 7 of

8 It is required to prove CLAIM (k+) is true ie ( k + ) = ( k + ) LHS = k + = k = 4 k = 4( ) + = 4 l + usig () = ( 4l + ) eacher s Markig = herefore CLAIM (k+) is true herefore by the riciple of Matheatical Iductio CLAIM () is true Guide/Aswers ie ( ) (g) Sice the su of the two itegers ad is eve + = k k = + = k = ( k ) = ie the differece of the two itegers is eve (h) = ( + )( ) For > + > ad > for > = rs is coposite r s uestio 5 (0 arks) (a) a a = = ak = ak + ak for k k CLAIM(): a = herefore a is odd ad CLAIM() is true CLAIM(): a = herefore a is odd ad CLAIM() is true Assue CLAIM(k) CLAIM( k )CLAIM( k ) CLAIM() are true ie ak ak ak Ka are all odd ie a = l a = K l K () k k WCA-WUC-EXSF age 8 of

9 It is required to prove CLAIM (k+) is true ie that a k + is odd a a + a k + = k = l + = ( l + ) ( ) = l + 4 k usig () = p p ie a k + is odd herefore CLAIM (k+) is true herefore by the Strog riciple of Matheatical Iductio CLAIM () is true eacher s Markig (b) (i) d = gcd = = = Guide/Aswers 87 = + = + = d = 544 ad 00 are relatively prie (ii) = d = = = 4 = 5 ( 87 ) = 4 87 ( ) = = = ( ) 87 ( ) 544 = = 46 = 5 (c) (i) he quotiet reaider theore: If ad d > 0 are both itegers the there exist uique itegers q ad r such that = dq + r ad 0 r < d (ii) Applyig the quotiet-reaider theore q r = q + r 0 r < Sice r = 0 = q or = q + or = q + (iii) Cosider three cases If = = ( q) = 9q = ( q ) = k k q If = q + ( q + ) = 9q + 6q + = ( q + q) + = l + = l If = q + WCA-WUC-EXSF age 9 of

10 ( q + ) = 9q + q + 4 = ( q + 4q + ) + = + = herefore the square of ay iteger has the for k or k + for soe iteger k (d) he followig cogruece classes odulo 7 are equal: [ 5] = [ 7] = [ 0] [] = [ 7] [ ] = [ ] ad a b c d By the defiitio of cogruece odulo (e) Let eacher s Markig ab ac od ab ac a( b c) Sice gcd ( a ) = by Euclid s lea ( b c) Guide/Aswers b c od (f) here are 6 letters i the alphabet herefore there are 6 = 676 cobiatios of first ad last letters herefore by the pigeohole priciple the uber with sae cobiatio of first 700 ad last letter = = 676 (g) [] x = [] for x i 4 [][] 0 = [] 0 [][] = [] [][] = [ 0] [ ][ ] = [ ] herefore x = [] [ ] uestio 6 (0 arks) (a) Aswer the questios below for the followig graph G: v e v e 6 e e e e 4 e 5 e 0 e e 8 v 5 v 4 v 9 e 7 (i) e e e e (ii) v v (iii) e e e 8 e 9 e (iv) v e 6 v ; v e 7 v WCA-WUC-EXSF age 0 of

11 (v) e 4 ad e 5 ; e 8 ad e 9 (vi) 5 (vii) (viii) A subgraph of G with vertices ad edges: e 6 v e 4 e 5 eacher s Markig v (ix) v Guide/Aswers e v 5 e 8 v 4 e 0 v e 4 v (x) v 5 e v e v 4 e 8 v 5 e 9 v 4 e 0 v e v e v e 4 v e 6 v e 5 v e 7 v G has a Euleria path because it has exactly two vertices of odd degree viz v 5 ad v (b) (i) A siple graph with four vertices of degrees ad 4: his siple graph does ot exist If it existed the vertex of degree 4 would have to be coected by edges to 4 distict vertices other tha itself his cotradicts the assuptio that the graph has 4 vertices i total (ii) A siple graph with six edges ad all vertices of degree : (c) wo isoorphic graphs: e 4 v 4 v 4 v v e 4 e e e e 7 e 6 e 7 e 6 e v v e v e 5 e v 5 v e 5 v 5 (d) WCA-WUC-EXSF age of

12 Kruskal s algorith: Edge Weight Will addig edge ake circuit? v Actio take Cuulative Weight of Subgraph v No Added v v No Added 4 ( v 4 ) v 4 No Added 8 v v 4 4 Yes Not added v v 4 Yes Not added eacher s Markig Guide/Aswers A iiu spaig tree is: v v 4 v v 4 (e) K 4 v v v v 4 v 5 v 6 Let V = { v v } V = { v v 4 v 5 v 6 } K { V E} 4 = is coplete bipartite because V V = V V V = φ ad each vertex i V is coected to each vertex i V by exactly oe edge WCA-WUC-EXSF age of

13 eacher s Markig Guide/Aswers END OF EXAM WCA-WUC-EXSF age of