## Vineet Loomba is teaching live on Unacademy Plus

BOARDS PREPARATIONS MADE EASY RELATIONS AND BINARY OPERATIONS CBSE AND STATE BOARDS CLASS 12 PREPARED BY: ER. VINEET LOOMBA IITIAN IIT-IEE MENTOr

100% SUCCESS IN CLASS 12 BOARD EXAMS ABOUT ME B. Tech. From IIT Roorkee IIT-JEE Mentor Since 2010. Youtube, Founder @ vi neetloomba.com unacademy & Many of my students have scored 100/100 & Currently running my own Coaching Institute for JEE Main and Advanced Search vineet loomba unacadem "on G00GLE

BOARDS PREPARATIONS MADE EASY BINARY OPERATIONS CBSE AND STATE BOARDS CLASS 12 PREPARED BY: ER. VINEET LOOMBA IITIAN IIT-IEE MENTOr

100% SUCCESS IN CLASS 12 BOARD EXAMS Let* be the binary operation on H given by a* b-L. C. M of a and b. find (a) 20 16 (b) Is *commutative (c) Is* associative (d) Find the identity of* in N. EXAMPLE: 20 * 162 L. C.M of 20 and 16|-(ii) a * b = L.C.M of a and b = L.C.M of b and a =b*a = 80 SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS (iii) a * (b * c) = a * (LC.M of b and c) = L.C.M of (a and L.C.M of b and c) = L.C.M ofa, b and c Similarity (a * b) * c = L. C.M ofa, b, and c (iv) a * 1 = L.C.M of a and 1 =a SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS EXAMPLE: Consider the binary operation* on the set 11, 2,3, 4, 5] defined by a * b- min. la, b]. Write the operation table of the operation*. 4 2 2 2 4 4 SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS EXAMPLE: Consider the binary operations :R x R->R and o Rx R ->R defined as ab la -b and aob a for all a, be R. Show that , is commutative but not associative, o' is associative but not commutative For operation'*' For Operation 'o o:R R R st. aob a Commutativity Commutativity a, bER a*b =la-b aob -a and boab i.e., is commutative o' is not commutative. SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS Associativity Associativity: V a, b, cER (aob) oc - aoc-a ao(boc) aob - a (aob) oc-ao (boc) a,b, CER (a * b) * c |a-b]*c a* (b*c)a b c ' is associative But la-b-c a is not associative. SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS EXAMPLE: A binary operation * on the set (o, 1, 2, 3, 4, 5) is defined as: atb, if a tb <6 a +b - 6, ifa+b6 Show that zero is the identity for this operation and each element 'a' (except O) of the set is invertible with 6-a, being the inverse of"a". For Identity Element: Let a be an arbitrary element of set 10, 1, 2,3, 4,5) Now, 0+a=0+a=a ...(i) a0 0+a<6V aE 10,1, 2, 3, 4,5] Eq. () and ()a0 0a a Vae 10,1, 2, 3, 4,5 Hence, 0 is identity for binary operation* SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROoORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS For Inverse: Let a be an arbitrary element of set 10, 1, 2, 3, 4,5) Now a (6-a) = a + (6-1)-6 =a+6-a-6 - 0 (identity) a = (6 _ a) + a-6 -6-a+a - 6 - 0 (identity) Eq. (i) and (ii)-,0+ (6-a)-(6-a) #A-0 (identity) ae(0, 1, 2, 3, 4.3 Hence, each element 'a' of given set is invertible with inverse 6 - a. Also, (6-a) a(6-a) 2 6] SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS (ii) Let (x, y) be identity element for on A, Then (a, b)" (x y) = (a, b) (a + x, b+y)=(a, b) y-0 But (0,0)g A .. For *, there is no identity element. SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LoOMBA (IIT ROORKEE)