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Consider the following statements in respect of the relation R in the set IN of natural numbers defined by xRy if x2 - 5xy + 4y2 = 0 :1. R is reflexive2. R is symmetric3. R is transitiveWhich of the above statements is /are correct ?
Concept:1. Reflexive: Each element is related to itself.R is reflexive if for all x ? A, xRx.2. Symmetric: If any one element is related to any other element, then the second element is related to the first.R is Symmetric if for all x, y ? A, if xRy, then yRx.3. Transitive: If any one element is related to a second and that second element is related to a third, then the first element is related to the third.R is transitive if for all x, y, z ? A, if xRy and yRz, then xRz.4. R is an equivalence relation if A is nonempty and R is reflexive, symmetric, and transitive.Calculation:We have R = {(x,y): x2 - 5xy + 4y2 = 0, x,y ? N}.Statement I: ReflexiveIf y = x, then x2 - 5x2 + 4x2 = 0? (x,x) ? R.? R is reflexiveStatement II: SymmetricTake (4, 1) i.e. x = 4 and y = 1We have (4)2 ? 5(4)(1) + 4(1)2 = 16 ? 20 + 4 = 0? (4,1) ? R.Also (1)2? 5(1)(4) + 4(4)2 = 1 ? 20 + 64 = 45 ? 0? (1,4) ? R.? R is not symmetric.Statement III: Transitive(16,4) ? R because(16)2 ? 5(16)(4) + 4(4)2 = 256 ? 320 + 64 = 0Also (4,1) ? R because? (4)2 ? 5(4)(1) + 4(1)2 = 16 ? 20 + 4 = 0Now, (16,1) ? R if (16)2 ? 5(16)(1) + 4(1)2 = 0? 256 ? 80 + 4 = 48 ? 0, which is not so.? (16,4), (4,1) ? R and (16,1) ? R? R is not transitive.? R is only Reflexive.
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