Exercise # 2.1 3, 7, , 3, , -9, 1, Solution: natural numbers are 3, , -9, 1, 2.5, 3, , , -9, 1, 2 2.5, 3, , -9, 1, , -9, 1, 2.
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1 Chter Chter Syste of Rel uers Tertg Del frto: The del frto whh Gve fte uers of dgts ts del rt s lled tertg del frto. Reurrg ( o-tertg )Del frto: The del frto (No tertg) whh soe dgts re reeted g d g the se order ts del rt s lled reurrg del frto. No-reurrg No-tertg Del frto: The o-tertg del frto whh the dgts re ot reeted the se order ts del rt s lled o-tertg d o-reurrg frto. Rtol uers:a uer whh e eressed the for of Khld Mehood M-Phl Aled Mthets where Z d Irrtol uers: A uer whh ot re lled rtol uer. e eressed the for of where Z d re lled rrtol uer. Rel uers: Uo of set of rtol d rrtol uers s lled the set of rel uers. Eerse #. Cosder the uers Q. Fro the followg whh re whole uers? Soluto: whole uers re 9-9 Q. Fro the followg whh re tegers? Soluto: tegers re -9-9 Q. Fro followg whh re rrtol uers? Soluto: rrtol uers re Eerse #. Q. Fro followg whh re turl uer? Soluto: turl uers re -9 Q. Fro the followg whh re rtol uer Soluto: rtol uer re Q.Fro the followg Whh re rel uers Soluto: rel uer Q. Fro the followg whh re rtol uers ut ot tegers? Soluto: rtol uers ut ot tegers. Q8. Fro the followg whh re teger ut ot whole uers Sol: Iteger ut ot whole uers -9 Q9. For the followg whh re tegers ut ot turl uers? Sol: Iteger ut ot turl uers -9 Q. Whh re rel uers ut ot tegers?
2 Chter Sol: Rel uers ut ot tegers Q. Wrte the del reresetto of eh of the followg uers. 9 8 Soluto Gve Q. Det eh uer o uer le. Proertes of Rel Nuers:- Closure Proerty:- (). W.r.t. Addto For eh R R (). W.r.t. Multlto For eh R. R Couttve roerty:- (). W.r.t. Addto For eh R R (). W.r.t. Multlto For eh R.. R Assotve roerty: (). W.r.t. Addto For eh R (). W.r.t. Multlto For eh R.... Idetty: (). W.r.t. Addto There ests R suh tht (). W.r.t. Multlto There ests R suh tht.. Iverse: (). W.r.t. Addto Eerse #. For eh R the there ests eleet ( ) suh tht ( ) ( ) R (). W.r.t. Multlto For eh Rthe there ests eleet suh tht R.. Dstrutve roerty: (). Multlto over Addto For ll R.. (). Addto over Multlto For ll.. R Proertes of eulty of Rel uers: Refleve Proerty For ll R Syetr Proerty For ll R Trstve roerty For ll R Addtve Proerty For ll For ll R R d d d Multltve Proerty For ll R d.. For ll R d.. Cellto Proerty (). W.r.t. Addto For ll R d For ll R d (). W.r.t. Multlto For ll R.. d For ll R.. d Proertes of eulty of rel uers Trhotoy Proerty For y R ether or or Ether s lso wrtte s Arhde roerty For ll R the there ests turl uer suh tht Trstve roerty For ll R d For ll R d Addtve roerty (). For ll R For ll R ().For ll R Khld Mehood M-Phl Aled Mthets
3 For ll Chter R Khld Mehood M-Phl Aled Mthets d Multltve Proerty ().For ll d.. For ll R R d.. () For ll R d.. For ll R d.. Ieulty ultltve verse For ll If () (). Eerse. Q. Wrte e of roertes used followg eutos. Ltter reresets rel uers. ). +(+)=(+)+ Assotve lw of ddto ). (+)=+ Dstrutve Lw of ultlto over ddto ). +=+= Estee of ddtve detty v). Estee of ultltve verse Q. Wrte the ssg uer. ). Aswer ssg uer s ). so Aswer ssg uer s ). so Aswer ssg uer s v). Aswer ssg uer s 8 Q. Choose the orret oto ). ). 8 ). 8 ). 8 d). 8 ). I whh of the followg llustrtes the ssotve lw of ddto? ). ). ). d). ). Whh of the followg llustrtes the ssotve lw of ultlto? Eerse. ). ). ). d). 8 Q. Do ths wthout usg dstrutve roerty. ). 9 9 Soluto: Gve ) 8 8 Soluto: Gve ). 8 Soluto: Gve 8 8 v). 8 Soluto: Gve Eoet or de:if rel uer s ultled wth tself tes the the rodut wll e.slrly the rodut of rel uer wth tself tes wll e wrtte s..... ( tes ultlto of wth tself) Rdd: I Or Ide: I s the de. Or Ide Or Eoet: I Bse: I s lled the rdd. s lled eoet or de. s lled Bse & s eoet Eerse #. Q. Wrte dow the de d rdd for eh of the followg eressos. ). y Aswer: Here de ). Rdd y Aswer: Here de Rdd ). Aswer: Here de Rdd Q. Trsfor the followg rdl fors to eoetl fors. Do ot slfy ).
4 Chter Eoetl for = ). Eoetl for = ). 8 Eoetl for = 8 v). Eoetl for = v). Eoetl for = v). Eoetl for = Q. Trsfor the followg eoetl for of eresso to rdl for ). rdl for = ). rdl for = ). 8 rdl for = v). y rdl for = v). 8 y rdl for = v). rdl for = Q. Slfy ). soluto Gve ). 8 soluto Gve 8 Khld Mehood M-Phl Aled Mthets ). y y soluto Gve v). y Eerse #. y y y y y y y y soluto Gve y v). 8 soluto Gve 8 v). y soluto Gve y y y y y y y Lws of Eoets:. Su of owers:. ower of rodut. y. y. ower of ower. uotet of owers wth se se:. where where. If.. Power of frto y y
5 Chter. Rtol eoets If s y ostve turl uer N d the es th root of s or d s defed y. y. Khld Mehood M-Phl Aled Mthets y Eerse #. y y Q. Wrte se eoet d the vlue of the followg. ). 9 Bse = eoet = - 9 vlue = ). Bse = ). eoet = vlue = Bse = - eoet = vlue = Q. If deote the rel uers the slfy the followg. ). Soluto: Gve ). Soluto: Gve 8 9 ). Eerse #. Soluto: Gve v). Soluto: Gve v) 8 9 Soluto: Gve v). Soluto: Gve v). Soluto: Gve v). Soluto: Gve ). Soluto: Gve 9 Q. Slfy the followg ). Soluto: Gve
6 ).. Chter Soluto: Gve ).. d. d Soluto: Gve. d. d. d v). Soluto: Gve v)... d Soluto: Gve... Q. Slfy the followg suh wy tht swer should ot ot frtol or egtve eoets. ). 8 Soluto: Gve ). Eerse #. Soluto: Gve ) Soluto: Gve v)... r r r r Soluto: Gve.. r r.. r r r r r r.. r r r rr r r r rrr r rr.. r r.. r r Q. Prove tht Soluto: Tkg LHS Khld Mehood M-Phl Aled Mthets
7 Chter 8... Khld Mehood M-Phl Aled Mthets 8 9 = RHS Hee roved. Cole uers: I 9 Guss Ger Mthet gve the oet of Cole Nuer s or Z where s lled the rel rt d s lled gry rt. Use of ole uers: Cole uers ly very ortt role Mthets d see. The use of ole uers s dsesle hyss Aeroutl d Eletrl Egeerg eselly the lyss of Eletr ruts. Cojugte of Cole uers: If the ojugte of Z s defed s s otg y terhgg the sg of gry rt. Eulty of two ole uers: Let d e two ole uers f d d gry rts d ff rel rts Addto of two ole uers: Let d d e two ole uers. Addto of deoted y d defe s d d d Sutrto of two ole uers: Let d d e two ole uers. sutrto of deoted y d defe s d d d d Multlto of two ole uers: Let d d e two ole uers. Multlto of deoted y. d defe s. d d d d d d d d Dvso of two ole uers: Let Eerse #. d d ole uers. Dvso of deoted y d d defe s d d d d d d e two d d d d d d d Modulus or the Asolute vlue of Z: The Modulus or the Asolute vlue of Z s defed s Ele: Add ole uer Soluto: Gve to dd Now 8 Ele: sutrt fro 8 Sol: Gve to sutrt fro E:Let d the fd Soluto: Gve d Now Ele: Let d fd the uotet Soluto: Gve d Now to rtole deotor
8 Chter Eerse. Q. Add the followg ole uers. ). 89 Soluto; gve tht Now ). 8 9 Soluto; Gve tht Now ). 8 9 Soluto; se so gve ole uer eoes Now 8 v). Soluto; Gve Now Q. Sutrt. ). fro Soluto; gve tht Now 8 ). 9 fro 9 8 Soluto; gve tht ). fro 8 Soluto; Gve fro Now 8 v). fro Eerse. Soluto; fro Now Q. Multly the followg ole uers. ). 8 Soluto: Gve Now 9 ). Soluto: Gve 8 Now 8 8 ). Soluto: Gve Now v). Soluto: Gve Now Q. Dvde the frst ole uer y seod. ). Soluto: gve Now To Rtole the deotor Multly & dvdg y ojugte of.e. 9 9 Khld Mehood M-Phl Aled Mthets
9 Chter ). Soluto: Gve Now To Rtole the deotor Multly d dvdg y ojugte of.e. Q. Perfor the dted oertos d redue to the for ). Soluto: Gve ). Soluto: Gve ). Soluto: Gve v). Soluto: Gve To Rtole the deotor & y ojugte of.e. 8 8 Revew Eerse 8 Q. Fd the ole ojugte of the followg ole uers. 8 ). Soluto: Gve ). Soluto: Gve 9 9 ). 8 9 Soluto: Gve v). Soluto: Gve Revew Eerse Q. Tell whether the followg re true or flse ). ). ). v). 9 Flse True Flse Flse Q. Selet the orret swer. ). The ddtve verse of ). ). ). ). s d). - e the roerty ). outtve ). Assotve ). Dstrutve d). losure ). ). ). ). d). v). Whh of the followg reresets uer greter th - ut less the? ). : ). : ). : : d). d v). f 8 8 ). ). - ). ). 8 = v). the =? ). ). - ). d). v). The ultltve detty of rel uers s ). ). ). - d). R 8 Khld Mehood M-Phl Aled Mthets
10 Chter v). s ). ostve teger ). egtve teger ). ether ostve or egtve d). Not teger ). For the f ). ). - ). d). ). ). ). ). - d). ot defed ). 9? ote : ). 9 ). ). 9 d). 9 9 Q. Slfy eh of the followg. ). Soluto: Gve ). Soluto: Gve ) Soluto: Gve 8 v). 9 Soluto: Gve 9 9 Q. Multly 8 8 Soluto: Gve to ultly Q. Dvde y Soluto: Gve to dvde y? Revew Eerse To Rtole the deotor Multly &dvdg y ojugte of.e. Q. Ne the roerty used Aswer: Multltve verse Q. Use lws of eoets to slfy Soluto: Gve Khld Mehood M-Phl Aled Mthets
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