VENN DIAGRAMS SUBSETS INTERSECTION. 108 SETS AND VENN DIAGRAMS (Chapter 5)

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1 08 SETS ND VENN DIGRMS (Chpter 5) 5 Suppose = fx j x<5, x 2 Z g, R = fx j 5 x 8, x 2 Z g, nd T = f, 7, 0, g. List the elements of: i ii R iii R 0 iv T 0 v R 0 [ T vi R \ T 0 Find: i n(r 0 [ T ) ii n(r \ T 0 ). True or flse? If n() =x nd n() =y then n( 0 )=y x. If µ then 0 = fx j x=2, x 2 g E VENN DIGRMS n lterntive wy of representing sets is to use Venn digrm. Venn digrm onsists of universl set represented y retngle, nd sets within it tht re generlly represented y irles. For exmple: ' is Venn digrm whih shows set within the universl set. 0, the omplement of, is the shded region outside the irle. Suppose = f,,,, 9g, = f,, 9g nd 0 =(, g. We n represent these sets y: SSETS ' 9 If µ then every element of is lso in. The irle representing is pled within the irle representing. µ INTERSECTION \ nd. onsists of ll elements ommon to oth \ It is the shded region where the irles representing nd overlp.

2 SETS ND VENN DIGRMS (Chpter 5) 09 NION [ onsists of ll elements in or or oth. [ It is the shded region whih inludes everywhere in either irle. DISJOINT OR MTLLY EXCLSIVE SETS Disjoint sets do not hve ommon elements. They re represented y non-overlpping irles. If the sets re disjoint nd exhustive then = 0 nd [ =. We n represent this sitution without using irles s shown. Exmple 5 Suppose we re rolling die, so the universl set = f, 2,,, 5, g. Illustrte on Venn digrm the sets: = f, 2g nd = f,, g = f,, 5g nd = f, 5g = f2,, g nd = f, 5g d = f,, 5g nd = f2,, g \ = fg \ = f, 5g, µ nd re disjoint ut [ =. d nd re disjoint nd exhustive

3 0 SETS ND VENN DIGRMS (Chpter 5) EXERCISE 5E. Consider the universl set = f0,, 2,,, 5,, 7, 8, 9g. Illustrte on Venn digrm the sets: = f2,, 5, 7g nd = f, 2,,, 7, 8g = f2,, 5, 7g nd = f,, 8, 9g = f,, 5,, 7, 8g nd = f,, 8g d = f0,,, 7g nd = f0,, 2,,, 7, 9g 2 Suppose = f, 2,,, 5,..., 2g, = fftors of 8g nd = fprimes 2g. List the sets nd. Find \ nd [. Represent nd on Venn digrm. Suppose = fx j x 20, x 2 Z + g, R = fprimes less thn 20g nd S = fomposites less thn 20g. List the sets R nd S. Find R \ S nd R [ S. Represent R nd S on Venn digrm. List the memers of the set: d 0 d i g k j e 0 f \ g [ h ( [ ) 0 f h e VENN DIGRM REGIONS We n use shding to show vrious sets. For exmple, for two interseting sets, we hve: is shded \ is shded 0 is shded \ 0 is shded Exmple On seprte Venn digrms shde these regions for two overlpping sets nd : [ 0 \ ( \ ) 0 [ mens in,, or oth. 0 \ mens outside interseted with. ( \ ) 0 mens outside the intersetion of nd.

4 SETS ND VENN DIGRMS (Chpter 5) Clik on the ion to prtise shding regions representing vrious susets. If you re orret you will e informed of this. The demonstrtion inludes two nd three interseting sets. DEMO EXERCISE 5E.2 On seprte Venn digrms, shde: \ \ 0 0 [ d [ 0 e ( \ ) 0 f ( [ ) 0 PRINTLE VENN DIGRMS 2 nd re two disjoint sets. Shde on seprte Venn digrms: 0 d 0 e \ f [ g 0 \ h [ 0 i ( \ ) 0 In the given Venn digrm, µ. Shde on seprte Venn digrms: 0 d 0 e \ f [ g 0 \ h [ 0 i ( \ ) 0 NMERS IN REGIONS Consider the Venn digrm for two interseting sets nd. This Venn digrm hs four regions: is in, ut not in is in ut not in is in oth nd is neither in nor in

5 2 SETS ND VENN DIGRMS (Chpter 5) Exmple 7 P ( ) () () Q ( ) If (5) mens tht there re 5 elements in the set P \ Q, how mny elements re there in: P Q 0 P [ Q d P, ut not Q e Q, ut not P f neither P nor Q? n(p ) = = n(q 0 )=8+2=0 n(p [ Q) =8+5+9 =22 d n(p, ut not Q) =8 e n(q, ut not P )=9 f n(neither P nor Q) =2 Exmple 8 Given n() =0, n() =2, n() =27 nd n( \ ) =, find: n( [ ) n( \ 0 ): Solving these, = ) =, =, d =2 () () () ( d ) n( [ ) = + + =8 We see tht = fs n( \ ) =g + =2 fs n() =2g + =27 fs n() =27g d =0 fs n() =0g n( \ 0 )= = EXERCISE 5E. If () mens tht there re elements in the set \, give the numer of elements in: () ( ) () () 0 [ d, ut not e, ut not f neither nor 2 Give the numer of elements in: X Y X 0 X \ Y ( ) ( ) ( ) () X [ Y d X \ Y 0 e Y \ X 0 f X 0 \ Y 0

6 SETS ND VENN DIGRMS (Chpter 5) () mens tht there re elements in the shded region. P Q Find: () () () () d n(q) n(p 0 ) n(p \ Q) d n(p [ Q) e n((p \ Q) 0 ) f n((p [ Q) 0 ) The Venn digrm shows us tht n(p \ Q) = nd P Q n(p )= + =. ( ) () ( ) ( ) Find: i n(q) ii n(p [ Q) iii n(q 0 ) iv n() Find if n() =. 5 For the given Venn digrm: () () () () d i find n( [ ) ii find n()+n() n( \ ) Wht hve you proved in? Given n() =20, n() =8, n() =9 nd n( \ ) =2, find: n( [ ) n( \ 0 ) 7 Given n() =5, n(n) =, n(n \ R) =, n(n [ R) =, find: n(r) n((n [ R) 0 ) 8 This Venn digrm ontins the sets, nd C. Find: n() n() ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) C n(c) d n( \ ) e n( [ C) f n( \ \ C) g n( [ [ C) h n(( [ ) \ C) PRINTLE PGE 9 On seprte Venn digrms, shde C 0 d [ e \ C f \ \ C g ( [ ) 0 h 0 [ ( \ C) C

7 SETS ND VENN DIGRMS (Chpter 5) Exmple 9 tennis lu hs 2 memers. 25 hve fir hir, 9 hve lue eyes nd 0 hve oth fir hir nd lue eyes. Ple this informtion on Venn digrm. Find the numer of memers with: i fir hir or lue eyes ii lue eyes, ut not fir hir. Let F represent the fir hir set nd represent the lue eyes set. F F () () () ( ) ( ) () ( d ) (8) d =2 + =25 + =9 =0 ) =5, =9, d =8 i n(f [ ) = = ii n( \ F 0 )=9 0 In Jmes prtment lok there re 27 prtments with dog, with t, nd 7 with dog nd t. 5 prtments hve neither t nor dog. Ple the informtion on Venn digrm. How mny prtments re there in the lok? How mny prtments ontin: i dog ut not t ii t ut not dog iii no dogs iv no ts? riding lu hs 28 riders, 5 of whom ride dressge nd 2 of whom showjump. ll ut 2 of the riders do t lest one of these disiplines. How mny memers: ride dressge only only showjump ride dressge nd showjump? 2 % of people in town ride iyle nd 5% ride motor sooter. % ride neither iyle nor sooter. Illustrte this informtion on Venn digrm. How mny people ride: i oth iyle nd sooter ii t lest one of iyle or sooter iii iyle only iv extly one of iyle or sooter? ookstore sells ooks, mgzines nd newsppers. Their sles reords indite tht 0% of ustomers uy ooks, % uy mgzines, 0% uy newsppers, % uy ooks nd mgzines, 9% uy mgzines nd newsppers, 7% uy ooks nd newsppers, nd % uy ll three.

8 SETS ND VENN DIGRMS (Chpter 5) 5 Illustrte this informtion on Venn digrm like the one shown. Wht perentge of ustomers uy: i ooks only ii ooks or newsppers iii ooks ut not mgzines? N M REVIEW SET 5 Consider = fx j <x 8, x 2 Z g. List the elements of. Is 2? Find n(). 2 Write in set uilder nottion: the set of ll integers greter thn 0 the set of ll rtionls etween 2 nd. If M = f x 9, x 2 Z + g, find ll susets of M whose elements when multiplied give result of 5. If = fx j 5 x 5, x 2 Z g, = f 2, 0,,, 5g nd = f,, 0, 2,, g, list the elements of: 0 \ [ d 0 \ e [ 0 5 List the sets: R S i R ii S iii S 0 d j g iv R \ S v R [ S vi (R [ S) 0 i f h e Find n(r [ S): In the swimming pool, people n swim utterfly nd n swim freestyle. The people who n swim utterfly re suset of those who n swim freestyle. There re 5 people in the pool in totl. Disply this informtion on Venn digrm. Hene, find the numer of people who n swim: i freestyle ut not utterfly ii neither stroke. 7 List the memers of set: d g i ii iii True or flse? f i µ ii \ = e iii [ = iv d=2 8 The numers in the rkets indite the numer of elements in tht region of the Venn digrm. Wht is the gretest possile vlue of n()? ( x ) () x (2- x) ()