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B RaryelR> $345,694 Codawain ~ Secovet $ et =8 The fulfillment of your dreams lies within you. Samos Bats 7 SUNIL SHARMA MATHEMATICS CLASSES ; RELATIONS (Ol. Lets Pallitei : Ql. S be the set of all triangles in a plane and let R be a relation in A, defined by bs ((A),A5): A, = 45}. Show that R is an equivalence relation in A. ; Let » be the set of all lines in xy-plane and let R be a relation in A, defined by R= de pleas LiLo}. Show that R is an equivalence relation in A. ; Find the set of all lines related to the line y= 3x + 5. Let Z be the set of all integers R be a relation in Z, defined as R a {(a, b) : a, b eZ , and (a—b ) is divisible by 5}. Prove that R is an equivalence relation. Let Z be the set of all integers and let R be a relation in Z, defined by R= {(a, b) : (a—b) is even}. Show that R is an equivalence wallet fi HL, Prove that the relation R in the set A {1, 2, 3, 4, 5} given by R= {(a, b): la-b lis even}, is an equivalence relation. Show that the relation R in the set A = {x: x € Z, 0 < x12} given by R= {(a, b); |a—b lis divisible by 4} is an equivalence relation. Find the set of all elements related to 1.— n me TER iwapow tert Let , be the set of all lines in a plane and let R be a relation in A, defined by R= ((L;,L2):L, £ L,}. Show that R is symmetric but neither reflexive nor transitive. (Q8._ Let N be thy set of all natural numbers and let R be a relation in N, defined by - R= t(a,b) ; ais a factor roy. 1men show that lk. retlexive and transitive ‘uc ndt ay symmetric. y Oe ‘\Q9. Let N be the set of all natural numbers and let R be a relation in N, defined by ~=° "6 R= {(a, b): ais a multiple of b}. Show that R is reflexive and transitive but not symmetric. Q10. Let S be the set of all real numbers and let R be a relation in S, defined by J R= {(a, b): (Itab) > 0}. Show that R is reflexive and symmetric but not transitive. QI1. Let S be the set ofall real numbers and let R be a relation in S, defined by R= {(a, b): as b}. Show that R is reflexive and transitive but not symmetric. O12. Let S be the set of all real numbers and let R be a relation in S, defined by NYY R= {(a,b): as b>}. Show that R satisfies none of reflexivity, symmetry and transitivity. Q13. Let S be the set of all real numbers and let R be a relation in S, defined by R= {(a, b): as b’}. Show that R satisfies none of reflexivity, symmetry and transitivity. b: Let N be the set of all natural numbers and let R be a relation on N xN, defined by (a,b) R(c, d) = ad = be. Show that R is an equivalence relation. 15. Let N denote the set of all natural numbers and R be the relation on NxN defined by (a, b) Ric, d) ifad(b + c) = be(a + d). Show that R is an equivalence relation. Ge eA fl, 2, 3------:: 9,} and R be the relation in AXA defined by (a, b) R (¢, 4). Ifat+d=b+c for (a, b), (c, d) in AxA, prove that R is an equivalence relation. 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