“Start by doing what’s necessary; then do what’s possible; and suddenly you are doing the impossible.” —*Francis of Assisi*

**Question 2.** Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y) : x and y have same number of pages} is an equivalence relation.

**Question 3.** Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

**Question 4.** Show that the relation R in the set {1, 2, 3} given by R = { (1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

**Question 5.** Show that each of the relation R in the set A , given by A. R = {(a, b) : |a – b| is a multiple of 4} B. R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.

**Question 6.** Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as center.

**Question 7.** Show that the relation R defined in the set A of all triangles as is equivalence relation. Consider three right angle triangles with sides 3, 4, 5, with sides 5, 12, 13 and with sides 6, 8, 10. Which triangles among are related?