Relation Math Question

Question

Let R = {(a,b)∣a divides b} be a relation on the set {Z}^+

Which of the following statements are correct? Tick all the corresponding answers.

a) R ⊆ Z^+ × Z^+

b) R is reflexive.

c) R is antisymmetric.

d) R is total.

e) R is a partial order.

Answer 1

a) Yes, of course. Z⁺ X Z⁺ covers all pairs (a,b), where a,b are in Z⁺

b) Yes. any number in Z⁺ divides itself

c) Yes. If a,b are in Z⁺ , a divides b AND b divides a => a = b

d) Yes. R is total because there is b in Z⁺ for every a in Z⁺. Namely, a. That is, in plain English, for every positive integer, there exists a positive integer which can be divided by it.

e) Yes. R is a partial order if it is reflexive, antisymmetric, and transitive. R is transitive because if a divides b, and b divides c, a divides c. Again, a,b,c are in Z⁺

1. Jason