RULES FOR MANIPULATING SURDS b. This is the addition law of surds with the same radicals. (ii)

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1 SURDS Defiitio : Ay umer which c e expressed s quotiet m of two itegers ( 0 ), is clled rtiol umer. Ay rel umer which is ot rtiol is clled irrtiol. Irrtiol umers which re i the form of roots re clled surds. For exmple,,,, d re irrtiol umers while 6, 8 d c e expressed i rtiol form. Defiitio : A geerl surd is irrtiol umer of the form, where is rtiol umer d is irrtiol umer, while is clled rdicl. RULES FOR MANIPULATING SURDS (i) c ( c). This is the dditio lw of surds with the sme rdicls. (ii) (iii) (iv) (v) (vi) (vii) d c d ( c).. d ( ).( c d ) c d. ( ) ( c d ) ( ) (viii) ( ) (ix) m m (x) m m c d. This is the sutrctio lw of surds with the sme rdicls. Simplifictio of surds Exmple; Simplify the followig (i) 7 (ii) 80 (iii) 8 (iv) 60 Solutio: Usig rule (i) 7 =. (ii) 80 = 6 6. (iii) 8 = 9 9. (iv) 60 =. Additio d sutrctio of surds Exmple; Simplify the followig (i) 0 8 (ii) 80 0 (iii) 8 6 Solutio: Usig rule d

2 (i) (ii) (iii) Rtioliztio of surds A surd of the form cot e simplified, ut we multiply the umertor d deomitor of process is clled rtioliztio. Useful hits o rtioliztio of surds (i). (ii) (iii) (iv) ( )( ) ( x y )( x y ) x y ( x y )( x y ) x y (v) the cojugte of is Exmple; (i) Rtiolize figure (iii) Express (iv) Express Solutio: (i) ( 8 6 i the form ) c e writte i more coveiet form. The, y. Such tht. This (ii) if. 7, fid the vlue of correct to sigifict i the form m where m d re rtiol umers. c, where d re rtiol umers

3 .7 (ii). 6 correct to sig.figure (iii) 6 7 m 8 6 d (iv) ( 8 ) ,, 8 8, c 8 (6 ) (6 )(6 ) Equtios ivolvig surds Exmple; (i) Solve the equtio ( x ) ( x ) (ii) simplify 6 (iii) Evlute 9 Solutio: (i) ( x ) ( x ) (x ) ( x ) () squrig oth sides of (), we hve x [ ( x )] x ( x ) x x x x x ( x ) x ( x ) squrig oth sides of () gi yields ( x ) x x x 0 x 0 or x if x 0, (x ) ( x ) ( ot solutio) if x, (x ) ( x ) x is the solutio () (ii) Let 6 x y () Squrig oth sides of (), we oti 6 x y xy x y, 6 xy By ispectio, x, y

4 6 (iii) Let 9 x y () The cojugte of () is 9 x y () Squrig oth sides of (), we hve: 9 xy x y 9 () Multiplyig () d (), we oti 9x 9 6. x y 7 () From () d () x y 9 x y 7 x 6 x 8, y EXERCISES () (i) 0 (ii) 98 (iii) 7 (vii) 7 (vii) (viii) (iv) 7 (v) (vi) 7 () If, Fid the vlue of () Give,, fid () Fid the positive squre roots of the followig : (i) 9 6 (ii) 7 () If x, express x x i its simplest form.

5 INDICES Defiitio : The product of umer with itself clled the secod power of the umer, while the umer, while its triple product is clled third power of the umer d its m fctors product is clled m mth power of the umer e.g. x, xx, xx... xm Defiitio : The umer which expresses the power is clled the idex or the expoet of the power of umer e.g The idex of The idex of The idex of m m RULES OR LAWS OF INDICES Give two positive itegers m, such tht m. () () m x m x m Sice xx... xm x xx... x m xx... x m xxx...( m ) m LOGARITHMIC EQUATIONS Exmple : Solve the equtios (i) x x 9 x x x x x x x 8 0 ( x )( x ) 0 x or x Exmple : Solve the equtios (ii) x x Tkig the log; of oth sides x x log log 0 0 x 0 x 0

6 x x 0 x x x () ( m m ) m m m ( ) x x... x ( xx... xm) x( xx... xm)... times. xx... xm m v x x... Similrly, m m m m () () 0, If m m m Exmples: Evlute (i) ( 8) (ii) (6) Solutio (i) ( 8) (8 ( ) 7

7 (ii) (6) Exercises ( ) () Show tht x () Evlute (Godm). x x LOGARITHMS Defiitio: The logrithm of tve o N to the se is defied s the power of which is equl to N, such tht if x N x N 0 Sice d 0 () log ( AB) A B A () log A B B B () log ( A ) B A Exmple: Evlute: (i) 9 (ii) 6 (iii) 9 9 (iv) 6, 6 Exmple: Use the tle to evlute: 06.0 (i) LAWS OF LOGARITHMS

8 Sice from the trsformtio rule 9 N 0 6 N y If y N, N y N ( ) N If we put N i (*) y N (**) N Aother form of (*) is N Exmple: Show tht: x ( x x ) ( ) Solutio: y x ( x ) ( ) x ( ) x ( )

9 SET THEORY Defiitio: A set is collectio of ojects or thigs tht is well defied. Here re some exmples of sets:. A collectio of studets i form oe. Letters of the lphet. The umers,,, 7, d. A collectio of ll positive umers. The cotet of ldy s purse The cocept of set is very importt ecuse set is ow used s the officil mthemticl lguge. A good kowledge of the cocept of set is, therefore, ecessry if mthemtics is to e meigful to its users. Nottio A set is usully deoted y cpitl letters; while the ojects comprisig the set re writte with smll letters. These ojects re clled memers or elemets of set. For exmple set A hs memers,, c, d. Covetio The listig of set A s,, c, d, s see ove is ot cceptle mthemticl specifictio of set. The correct represettio of set tht is listed is to write the elemets, seprted y comms d eclosed etwee rces or curly rckets. e.g., set,, c, d. A. The sttemet is elemet or memer of set A or elogs to is writte i the mer A. The cotrry sttemet tht does ot elog to A is writte s: A. There re two wys of specifyig set. Oe wy is y listig the elemets i the set, such s: A,, c, d.. A secod wy of specifyig set is y sttig the rule or property which chrcterizes the set. For exmple, B x / x. or B x / x. iterchgely, with ech s such tht. The represettio, x / x.. Notice, the stroke/or colo: c e used B is red s follows: B is set cosider of elemets x, such tht is less th x d x is less th.

10 If set is specified y listig its elemets, we cll it the tulr form of set; d if it is specified y sttig its property, such s C x / x is odd, the it is clled the set uilder form. Fiite d Ifiite Sets A fiite set is oe whose memers re coutle: for exmple, the set of studets i Form. Other exmples re: (i) the cotets of ldy s hd-g; (ii) whole umers lyig etwee d 0; (iii) memers of footll tem. The fiite set is itself i exhustive; reders c give other exmples of fiite set. A ifiite set is oe whose elemets re ucoutle, s they re ifiitely umerous. Here re few exmples of the ifiite sets; (i) Rel umers. (ii) Rtiol umers (iii)positive eve umers (iv) Complex umers The mi distictio etwee fiite set d tht fiite set hs defiite egiig d defiite ed, while the ifiite set my hve egiig d o ed or vice vers or my ot hve oth egiig d ed. For exmple, we specify the set of positive eve umers, s follows: P,,6,... or P x : x, x is eve The set of rel whole umers which ed with the umer is writte s follows: SUBSETS Q, the we sy Q is cotied i P, d we use symol ' ' to deote the sttemet is cotied i, or is suset of. Thus Q P, is redy s ' Q is cotied i P '. More ptly put, Q is cotied i P if there is x, such tht x Q implies x Q. The sttemet Q is cotied i P c e put i reverse order s ' P cotis Q' d we write P Q. However, this form is ot very populr. If Q is ot suset of set R,,, the we write Q R. It should e oted tht uless every memer or Q is lso memer of P, the c we sy Q is suset of P. Suppose P,, c, d, e, f d c, d, e.

11 EQUITY OF SETS Two sets of X d Y re equl if d oly if X Y d X X,, d Y,, d X Y. Note tht the rerrgemet of the elemets if set does ot lter the set. TYPES OF SETS Null or Empty Sets Y. Suppose Null mes void, therefore, ull set is empty set, or set tht hs o memers. The ull set is deoted y the symol. Note tht 0 cot e clssified s ull set, ecuse it hs elemet, zero. Sigleto Ay set which hs oly oe memer is clled sigleto. e.g., The Uiversl Set is sigleto. Set is suset of lrger set is clled the uiversl set or empty, the Uiverse of Discourse. Thus, i y give cotext, the totl collectio of elemets uder discussio is clled the Uiversl set. The symol U or is ofte used to deote uiversl set. For exmple, if we toss die, oce, we expect to hve either,,,,, or 6, s ed result. If there re o other expected results differet from this umers, the we sy, for this prticulr experimet, the uiversl set is,,,,,6. Thus uiversl set is the totl popultio uder discussio. Proper Susets If P is suset of Q d if there is t lest oe memer of Q which is ot memer of P, the P is proper suset of Q d we write P Q. Cosider the set,,,,,, A,,,. The followig sets,,,,,,,,,,,,, re susets of A. The set,, proper suset of A ; wheres ll others icludig re proper susets of A. Thus,,,,, ut,,,,,,,. Power Set is ot The collectio of ll the susets of y set S is clled the power set of S. If set hs memers, where is fiite, the the totl umer of susets of S is. Occsiolly we deote the power set S y S.

12 For exmple: Let A,, c. The susets of A re,, c,,,, c,, c,,, The power set of A writte Exmple: Fid the power set () S,) S,,, () Solutio: p ( A) susets; s see ove. S of the sets () S,,,,, () P,,,,,,, c d. I this exmple, () cotis oly two elemets d, Ve Euler Digrms The theory of set c e etter uderstood if we mke use of the Ve-Euler digrms. The Ve-euler digrm is istructive illustrtio which depicts reltioship etwee sets. Suppose X Y d X Y, we c represet this sttemet i Ve-Euler digrm s follows: Y Y X X X A B

13 Set Opertios I set, we use the symols red uios d red itersectio s opertios. These opertios re similr ut ot exctly the sme s the opertios i rithmetic. At the ed of this chpter, reder of this topic should e le to idetity res of logy etwee the opertio i rithmetic d those of set. Uio of Sets Defiitio The uio of sets A d B is the set of ll elemets which elogs to A or B or to oth A d B. This is usully writte s A B, d red ' A uio B '. I set lguge, we defie A B s: A B x : x A or B. The shded portios i the Ve-Euler digrm i A B A B A B The Itersectio of Sets The itersectio of sets AdB is the set of elemets which elog to oth AdB. Simply, ' A itersectio B' writte A B cosists of elemets which re commo to oth AdB. The Ve-Euler digrm which represet A B is shded portio. I set lguge A B x : x Adx B

14 Complemet of Sets The complemet of set x is the set of elemets which do ot elog to x, ut elog to the uiversl c. set. The complemet of set x is usully represeted y x' or x c The complemet of x ' or x. The complemet of x is represeted i the Ve-Euler digrm U x x I set lguge, A c x : x U, x A The Alger of Sets The opertios of uio re loosely logous to those of dditio d multiplictio i umer lger. By this toke we c pply the lws of lger coveietly to sets without loss of geerlity. The Closure property If X d Y re sets which re susets of the uiversl set U the the followig hold: X Y U d X Y U. The logy i umer lger; usig those opertios of d x re R d x 6R; where R is the rel umer system. If the dditio or multiplictio of d gives some umer tht cot e foud i the rel umer system R, we sy the opertio of or x is ot closed. Similrly i set theory, the opertios of uio d itersectio re closed.

15 The Commuttive Lw X Y Y U X Y Y X. Prllel exmples i rithmetic re d X X. Thus y two sets re commuttive with respect to d. The Associtive Lw X ( Y Z) ( X Y ) UZ d X ( Y Z) ( X Y ) Z Agi, sets oey the ssocitive lw. The Idetity I every dy rithmetic, 0 0 d X X, re two correct solutios. The zero, i the first cse is clled the dditive idetity; while i the secod cse is clled the multiplictive idetity. By similr logy, every set hs qutities d U with the property tht: (i) X X X (ii) X U U X X Thus, is the idetify with respect to uio d U is the idetity with respect to itersectio. Iverse I the set of rel umer R, ( ) ( ) 0 d X X. This umer x operted o its iverse gives idetity. i.e., x Iverse idetity. Similrly i set theory, every set hs iverse with respect to the opertios of d (i) X X ' X X ' U U d U U X U (iii) X X X ' X X X ' X ' d

16 The distriutive Lw X ( Y Z) ( X Y ) ( X Z) d X ( Y Z) ( X Y ) ( X Z) The opertio of uio is distriutive over the opertio of itersectio d vice vers. The Lws of complemettio (i) X X ' d (ii) ( X ' ) X (iii) ( X Y )' X Y ' (iv) X Y )' X ' Y ' (ii) (iv) re clled de Morg s Lws