Let T be the set of all triangles in a plane with R a relation in T given by` R ={(T _(1) , T _(2)): T _(1)` is congruent to `T _(2)`} Show that R is an equivalence relation.

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L _(1) , L _(2)) : L _(1) is parallel to L _(2)}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x +4.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L _(1), L _(2)) : L _(1) is perpendicular to L _(2) }. Show that R is symmetric but neither reflexive nor transitive.

Let T be the set of all trianges in the Euclidean plane and let a relation R on T be defined as a R b, if a is congruent to b, for all a, b in T Then R is

Show that the relation R defined in the set A of all triangles as R={(T _(1), T _(2))):T _(1) is similar to T _(2) } is equivalence relation. Consider three right angle triangles T_(1) with sides 3,4,5,T_(2) with sides 5,12,13 and T_(3) with sides 6,8,10. Which triangles among T_(1), T_(2) and T_(3) are related ?

Show that the relation R defined in the set A of all polygons as R = {(P _(1), P _(2)): P _(1) and P _(2) have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3,4 and 5 ?

Show that the relation R in the set of all integers, Z defined by R = {(a, b) : 2 "divides" a - b} is an equivalence relation.

If R_(1) and R_(2) are equivalence relations in a set A, show that R_(1) nn R_(2) is also an equivalence relation.

Let L be the set of all straight lines in the Euclidean plane. Two lines l_(1)" and "l_(2) are said to be related by the relation R" iff "l_(1) || l_(2) . Then the relation R is :

Let R be a relation on a set A such that R= R^(-1) . Then R is :