Exercise 1. Let R = {(x, y) | x, y E Z and x + y = 0, } be a relation on Z. Prove or disprove if R is (1) reflexive, (2) symmetric, and (3) transitive. II

Define a relation R on the x-y plane as follows: Given two points (x1 ,y1) and (x2 ,y2) in the x-y plane, (x1 ,y1) R (x2 ,y2) if and only if the distance between them is a whole number. Is R an equivalence relation? Prove or disprove

2. Define a relation R on Z as follows: a Rb if and only if 5 | (a² - b²). b) Find four integers which are equivalent to 3 (excluding 3 and -3).

Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R = IdA, then S is surjective. If the assertion is not true, find additional assertions that will make it true.