32) Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from the equivalence relation. (x1,y1)R(x2, y2) in RXR if x + y = x + y Here we are relating two ordered pairs defined by an equation. Similarly, using this equation: (x1,y1)R(x1,y1) if x + y = x + y. Also, (x2, y2) R(x1,y1) if x Also, (x2, y2) R(x3, y3) if x + y = x + y. + y = x + y. Reflexive: Yes. (x1,y1) R(x1,y1) since x + y = x + y must be true. Symmetric: Yes. Suppose x + y = x + y2, which implies (x1,y1)R(x2, y2). Reversing the equation, x + y = x + y. Hence, (x2, y2) R(x1,y1). (3 Transitive: Yes. Suppose x + y = x + y and x + y = x + y, which respectively imply (x1,y1)R(x2, y2) and (x2, y2) R(x3, Y3). (6) Solving simultaneous equations, x + y + x + y = x+y+x+y, implies x + y = x + y by the Cancellation Rule. Hence, (x1,y1) R(x3, Y3). Partition: {(a, b), (-a, b), (a, b), (-a, b), (a, a), (-a, -a), (a, -a), (-a, a)} for a R