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Page., , Group and Rings ===, , , , , , , , Binary operation —, , , , let § be any non-empty set. A function f: sxs35, , , , is known as a binary operation on S, , , A operation stor defined ons is said 4o be binary if, , , , axbes 3 Va,bes,, Ex., , , , 1) Let ‘N’ be a set of natural numbers and ‘+’ defined on s then, C+) is a binary operation on N because addition of natural, , , , no. is hatural ., , , , i.e. GtbEN, Va,ben., , , , 2) If we define subtraction C) on the set of natural no. N, , then subtraction is not binary operation on ‘N -because subtra , , , ction of +wo natural no. is not natural no, , , ie. a-b¢@N, Va,ben., , , , (2-3 =-1) EN, , , , , , 3) Usual product CG) is.a binary operation: on N,W+Z,9,R4C., , , Similarly. asual addition C+) is also binary operation on, , , , N,W,Z,@.R &c. But, ©) is binary on 2,G,,.R¢C- Usual, , , , division of no. is a binary operation on @,R,C., , , , , , 4) a*® b= atb-2 3 Va,ben, , , , then * is not binary operation an N., , , , xt = 141-2 EN, , , , Bud, 14 =141-1 then X is binary aperafon on N but pot, , , , binary aperation on W. (Since 0¥0 = 0+0-1=-1 ¢WN), , , , , , Group —, , , , Let G be any non-empty set & * bea binary operation, , , , , , define on G: Get :G together with» * satisfies following, , , , , , , , conditions: . a, , , , , , , , Scanned with CamScanner
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Ee, , , , Examples 8—, , , , , , Show that set of integers Z is an abelian group w.rt. ysual, +., , , , Let z be the set of integers and usual addition is q, , , , binary operation define on Z., , , , , , To prove that Z is an abelian group wort. ‘+’., |, i) Closure property —, | We know that, addition of two integers is an integer., ‘-athez' 3 ¥Yabez, | z is closed wort. ‘+’. a _, ii) Associativity :, We know that, addition of integers is an associative —, (a+b) +c = a+ (bie); Vasbsc EZ, 2 is associative w.rt.‘+’., , , , li) Existence of identity:, , There exiet an element 0€.2 st., ato = 9 = O+0 3 VGEZ. ,, » £0’ is identity element of z wrt. ‘+’., , , , |, liv) Existence of inverse :, For dny 9EZ +here exists. "~a’ € z ‘alt., , , , a+t(-a)=0=Ca)ta 3 Yaez, , , , This shows that ‘-a’ is inverse element of ain z, , , rr, , , , This all conditions of group satiety by the set Zwrt iy, , , , 3 , Heit:, , , , _z fs qroup wart., , , , , , , , , , , , , , , , Scanned with CamScanner
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Date, Page., , , , , , , , , , a, Sl Hes identity element of 9 * w.r.t. ysual product., , , , , , iv) Existence of: inverse —, , , , For any ae@* 4 t/q €9* st., , , , q: =it=i1.q, , , , 1, qd q, , , , This shows that 4/q is inverse element of 9* wrt., , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , r usual product., All conditions of group are satisfied by @* wrt., usual product., *. Q* ig group w.r.t. usual product., v) Commutative :, , | We know that, product of two rational no. js, | commutative . aches, J i.e. ab =ba 3 Va,beQ*., , @* is an abelian group w.r+. usual product., , , , Scanned with CamScanner
