-D is Reflexive -D is not symmetric -D is transitive
Work Step by Step
--D is reﬂexive: -[We must show that for all positive integers m,m D m.]Suppose m is any positive integer. Since m = m·1, by deﬁnition of divisibility m | m. Hence m D m by deﬁnition of D. -- D is not symmetric: - For D to be symmetric would mean that for all positive integers m and n, if m D n then n D m . By deﬁnition of divisibility, this would mean that for all positive integers m and n, if m|n then n|m. But this is false. As a counterexample, take m =2 and n =4. Then m|n because 2|4 but n (not|)m because 4(not|2). -- D is transitive: - To prove transitivity of D, we must show that for all positive integers m,n, and p, if m D n and n D p then m D p. By deﬁnition of D, this means that for all positive integers m,n, and p, if m|n and n | p then m | p.