Design maintenanceand reliability of engineering systems: a probability based approach

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1 Desg mateaead relablty of egeerg systems: a probablty based approah CHPTER 2. BSIC SET THEORY 2.1 Bas deftos Sets are the bass o whh moder probablty theory s defed. set s a well-defed olleto of objets. The objets are alled elemets or members of the set. Typally a set s deoted by upperase letters, B, C, P et. ad the elemets are deoted by lowerase letters a,b,, x, y et. set s ompletely desrbed by ts members. The desrpto a be aheved ether by () lstg (.e., eumeratg) the members, e.g.: X = {a,e,,o,u} whe desrbg the set of vowels. or, () by statg the membershp rule, e.g.: X={x: x s a teger betwee 1 ad 100} whe desrbg the set of the frst 100 atural umbers. The seod approah s more powerful. Symbolally, x states x s a elemet of, ad x deotes otherwse. Uversal set: I the otext of a problem, all sets of terest may be subsets of some large fxed set. Ths superset s alled the uversal set. Null set: The ull set, or the empty set, s the set wh o elemets. It s deoted by the speal symbol. Coutable set: set s outable f ts members a be plaed a oe-to-oe orrespodee wth the set of atural umbers. Otherwse, the set s uoutable. 2.2 Set relatos Subset: If every elemet of a set s also a elemet of set B, the s alled a subset of B, wrtte symbolally as: B,orB. If s a subset of B ad B has at least oe elemet that does ot belog to, the s a proper subset of B, wrtte symbolally as: B,orB. Superset: If B, the B s alled a superset of. If B, the B s a proper superset of. Equalty of sets: If every elemet of s a elemet of B ad ve versa,.e., Bad B, the the two sets are equal, wrtte as: = B. Trastvty: If Bad BC, the C. 2.3 Operatos o sets Ve dagrams a be used for graphal represetato of sets ad ther relatos. Operatos o oe or more sets produe ew sets. Bas operatos are: omplemetato, dffereg, symmetr dffereg, et.

2 Badurya Bhattaharya, IIT Kharagpur Boolea ombato of sets Gve two sets ad B, ther terseto C s the set suh that t otas oly those elemets that belog to both: C BC { x: xad x B} (2.1) The uo s the set D suh that t otas elemets that belog to or B or both: Idettes D B D{ x: xorxbor x B} (2.2) Varous equaltes a be desrbed through set operatos dempotet law, ommutatve law, assoatve law, dstrbutve law, voluto law, de Morga s law et. Idempotet laws (1 a ) (1 b ) ssoatve laws (2 a ) ( B) C ( B C) (2 b ) ( B) C ( B C) Commutatve laws (3 a ) B B (3 b ) B B Dstrbutve laws (4 a ) ( BC) ( B) ( C) (4 b ) ( BC) ( B) ( C) Idetty laws (5 a ) (5 b ) U (6 a ) U U (6 b ) (7) ( ) Ivoluto law (8 a ) U Complemet laws (8 b ) (9 a ) U (9 b ) U page 15 Last updated: 19-Feb-17

3 Desg mateaead relablty of egeerg systems: a probablty based approah DeMorga`s laws (10 a ) ( B) B (10 b ) ( B) B Partto of a set partto P= { } of the uversal set U s a olleto of mutually exlusve ad olletvely exhaustve sets, :, j j U (2.3) 2.4 Lmts of sets Feller IV.1: Reall lmt of the sequee of futos { f ( x )}. For a gve value of x (x s omtted to make the otato leaer), the term lm f f deotes the maxmum of the sequee of mma: lm f f max{m f, f,...} f (2.4) j j1 k 0j j1 kj The otato o the rght s the shorthad for max (deoted by up or uo) ad for m (deoted by ap or terseto), respetvely. Lkewse, the mmum of the sequee of maxma s: lm sup f m{max f, f,...} f (2.5) j j1 k 0 j j1 k j These oepts dretly apply to a sequee of sets { k }. The fmum ad supremum of the sequee are the sets defed respetvely as:.e., set of pots abset a lm f k w: (1- I k ( w)) (2.6) j1 kj k fte umber of k 's.e., set of pots preset lm sup k w: I k ( w) (2.7) j1 kj k all k 's The lmt of the sequee { k } exsts f the two lmts are equal ad may be deoted : lm f lm sup (2.8)

4 Badurya Bhattaharya, IIT Kharagpur whh may be wrtte short as: lm or (2.9) From P6, Resk Probablty Path: k let k 0, k 1 k B f k 0, 0, k k k1 1 k The, lm f 0, f 0, 0, 0,1 1 k 1 k k B sup k k 0, 0,1 k k k k 1 lm sup lm B lm 2.5 Ordered sets 0,1 0,1 ordered set S s a set whh a order s defed. order o a set S s a relato < wth the followg two propertes (Rud): 1. If x Sad y S the oe ad oly oe of the followg three statemets s true: x y x y y x (2.10) 2. If three elemets x,y ad z belog to S, ad f x y ad y z, the x z (2.11) The relato < may be read as s less tha or s smaller tha or preedes Supremum or least upper boud: S s a ordered set ad E S. If there exsts Ssuh that x for every x E, we say E s bouded above ad s a upper boud of E. Now, f s a upper boud of E suh that ay s ot a upper boud of E, the s alled the least upper boud or supremum of E: sup E (2.12) The supremum may or may ot belog to E. ordered set S has the l.u.b. property f for ay o-empty subset E that s bouded above, page 17 Last updated: 19-Feb-17

5 Desg mateaead relablty of egeerg systems: a probablty based approah ts supremum sup E exsts S Ifmum or greatest lower boud: Smlar to the sup defto above: Let G be a ordered set ad G S. If there exsts Ssuh that x for every x G, we say G s bouded below ad s a lower boud of G. Now, f s a lower boud of G suh that ay s ot a lower boud of G, the s alled the greatest lower boud or fmum of G: f G (2.13) The fmum may or may ot belog to G. ordered set S has the g.l.b. property f for ay o-empty subset G that s bouded below, ts fmum f G exsts S. 2.6 Set algebra Let be ay set 1. o-empty olleto of subsets of s a algebra of sets (.e., a feld) f: wheever 1, 2 are, so are \ 1 (.e., omplemet of 1 ) ad 1 2 (ad therefore 1 2 also). Geeralzg, f 1, 2,, ( fte) are, so are 1 U 2 U ad 1 2. Example: Let ={a,b,}. The we ould defe a feld as: =,, {a}, {b,} algebra (or feld or Borel feld): The algebra desrbed above s a algebra of sets f t holds for a outably fte 2 olleto 1, 2,. That s, wheever, the sequee 1, 2,, belogs to, so does 1. I other words, a algebra of subsets of a gve set otas the empty set ad s losed wth respet to omplemetato ad outable uos. 3 1 the otext of probablty, X s osdered to be the sample spae 2 fte sequee (of sze ) s a futo whose doma s the frst atural umbers. fte sequee s a futo whose doma s the set N of atural umbers. set s alled outable f t s the rage of some sequee (fte or fte). The set s fte ad outable f t s the rage of some fte sequee. The set s outably fte f t s the rage of some fte sequee. 3 Whe the evets 1, 2,, are outably fte, we a take the Borel feld osttutg the probablty spae to osst of all subsets of. Whe s uoutably fte (e.g., the real le R), we do ot wat the Borel feld to be the olleto of all subsets of to osttute a probablty spae. I ths ase, we oly osder evets of the type ad the Borel feld to osst of all fte tervals R. X x

6 Badurya Bhattaharya, IIT Kharagpur Measurable spae: ouple (, ) s a measurable spae where s ay set ad s a algebra of subsets of. subset of s measurable wth respet to f. Measure: measure m o a measurable spae (, ) s a o-egatve set futo defed for all sets of the algebra, f t has the propertes: () m() = 0. (2.14) () If 1, 2, s a sequee of dsjot sets of, the m( ) m( ). (2.15) 1 1 Measure spae: measure spae (,, m) meas a measurable spae (,) together wth a measure m defed o. Measurable futo: Let (, ) ad (, ) be two measurable spaes. The the futo (or map) f : ' s alled measurable f the verse satsfes f 1 ( ') (2.16) I the speal ase that s the sample spae ad the rage of f s the exteded real le,.e., ' ad ' B( ) the sgma algebra of tervals o the real le, the f must satsfy ay oe of the followg odtos order to be a measurable futo wth respet to : { x: f( x) } for eah { x: f( x) } for eah { x: f( x) } for eah { x: f( x) } for eah The, as we wll see CHPTER 4, f s alled a radom varable. page 19 Last updated: 19-Feb-17