Answer to Question #284206 in Discrete Mathematics for wajji

Question

Show that the relation R on Z × Z defined by (a, b) R (c, d) if and only if a + d = b + c is an equivalence relation. Note: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

Answer 1

reflexive: a + b = b + a, so (a, b) R (a,b)

symmetric: if a + d = b + c then c + b = d + a, so if (a, b) R (c, d) then (c, d) R (a, b)

transitive: if a + d = b + c and c + f = d + e then

$a + d+c+f = b + c+d+e\implies a + f = b + e$

so if (a, b) R (c, d) and (c, d) R (e, f) then (a, b) R (e, f)