Question

Let N+2 denote the natural numbers greater than or equal to 2. Let mRn if gcd(m, n) > 1. The binary relation R on N2 is

(a) Reflexive, Symmetric, Not Transitive
(b) Reflexive, Not Symmetric, Transitive
(c) Reflexive, Symmetric, Transitive
(d) Reflexive, Not Symmetric, Not Transitive (e) Not Reflexive, Symmetric, Not Transitive

Answer 1

a) Reflexive, Symmetric, Not Transitive

Explanation:

Given
`a\in N+2` we have that
`a\ge 2` and then
`\gcd(a,a)=a\ge 2` thus the binary relation is reflexive. To show that is symmetric note that for
`a,b\in N+2` we have
`\gcd(a,b)=\gcd(b,a)` which implies the symmetric. In fact, the relation is not transitive, for example note that
`\gcd(7,7^2)=7` and
`\gcd(5^2,5)` but
`\gcd(7,5)=1<2`.