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.5). Let R be a relation from a set A to another set B. Then R is of the form {(x, y): x ? A and y ? B}. The inverse relationship of R is denoted by R-1 and its formula is R-1 = {(y, x): y ? B and x ? A}.Calculation:Statement I: If R is reflexive, then R-1 is also reflexiveR is reflexive? (a,a) ? R, a ? A? (a,a) ? R?1 [by the definition of R-1]? R?1 is also reflexive relation.Statement II: If R is symmetric, then R-1 is also symmetricLet (b,a) ? R?1? (a,b) ? R, a,b ? A [by the definition of R]? (b,a) ? R [R is symmetric]? (a,b) ? R?1 [by the definition of R-1]If (b,a) ? R?1 then (a,b) ? R?1? R?1 is also symmetric relation.Statement III: If R is transitive, then R-1 is also transitiveLet (b,a), (a,c) ? R?1? (a,b), (c,a) ? R [by the definition of R-1]? (c,a),(a,b) ? R? (c,b) ? R [R is transitive]? (b,c) ? R-1 [by the definition of R-1]If (b,a), (a,c) ? R-1 then (b,c) ? R-1? R?1 is also transitive relation.? R?1 is reflexive, symmetric and transitive.