## 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 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 EXAMPLE: Let T be the set of all triangles in a plane with R a relation in T given by R = {(T1, T): T1 is congruent to Tz). Show that R is an equivalence relation. R is reflexive, since every is congruent to itself. (T1T2)E R similarly (T2T1) E R > since T1-T2 (TiT2) e R, and (T2,T3) e R (T1T3) E R Since three triangles are congruent to each other. SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS EXAMPLE: Let L be the set of all lines in plane and R be the relation in L define if R = {(11, L2 ): L1 is to L2). Show that R is symmetric but neither reflexive nor transitive. R is not reflexive, as a line L1 cannot be to itself i.e (L1,L) E R 3 2 Then L1 can never be to L3 in fact Lil L3 ie (L1,L2) E R, (L2,L) E R. But (L1, L3)E R SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS EXAMPLE: Show that the relation R in the set Z of integers given by R- ((a, b) : 2 divides a-b). R is reflexive, as 2 divide a-a 0 ((a,b)e R,(a-b) is divide by 2 > (b-a) is divide by 2 Hence (b,a) e R hence symmetric. Let a,b,c E Z If (a,b) ER And (b,c) e PR Then a-b and b-c is divided by 2 a-b +b-c is even (a-c is even (a,c) e R Hence it is transitive. SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS EXAMPLE: Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6) as R-((a, b): b a+1 is reflexive, symmetric or transitive Symmetric or transitive R is not reflective, because (1,1) e R R is not symmetric because (1,2)e R but (2,1) R (1,2) e Rand (2,3) E R But (1,3) R Hence it is not transitive SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS EXAMPLE: Let L be the set of all lines in Xy plane and R be the relation in L define as R = {(Li, L2): L1 ll L2} Show then R is on equivalence relation. Find the set of all lines related to the line Y=2x+4. LiIL e (Li, L ) e R Hence reflexive Lill L2 then L2 L i.e (LIL2)E R We know the L1IL2 and L21IL3 Then L1|| L3 (L2L)e R Hence symmetric Hence Transitive. y = 2x +K When K is real number SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS EXAMPLE: Show that the relation in the set R of real no. defined R [(a, b): as b3 ], is neither reflexive nor symmetric nor transitive. (i) (a, a) e R as asa' Which is false R is not reflexive. (ii) a sb' and bsa Which is false R is not symmetric. (ii) asb', bse' then asc4 Which is false SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)

100% SUCCESS IN CLASS 12 BOARD EXAMS EXAMPLE: Show that the relation R defined in the set A of all triangles as R = {(7,5) : Ti is similar to T is an equivalence relation. Consider three right angle triangles T with sides 3, 4, 5. T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T , T2 and T3 are relations? (Ti,T2) e F T is similar to T2 and T2 is similar to T3 Tl is similar to T2 T2 is similar to T1 T1 is similar to Ta (T2, Ti) e R R is symmetric .'. R is transitive. Hence R is equivalence SUCCESS IN MATHEMATICS CLASS 12 BOARDS ER. VINEET LOOMBA (IIT ROORKEE)