Question

Determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answers.Recall that a prime number is an integer that is greater than 1 and has no positive integer divisors other than 1 and itself. (In particular, 1 is not prime.) A relation P is defined on ℤ as follows:For every m, n is in ℤ, m P n ⇔ ∃ a prime number p such that p | m and p | n.

Answer 1

Answer:

Is Reflexive.

Is Symmetric

Is NOT transitive

Step-by-step explanation:

Is Reflexive.

Because any integer  $n$ is divided by some prime $p$ . Since $p | n , p|n$ it is true that  $n P n$

Is Symmetric

Suppose that     $n P m$    then  there is a $p$ such that  $p|n , \,\, p|m$, then  $p|m , \,\, p|n$   and $m P n$.

Is NOT transitive

Notice that   $3P6$   and    $6P8$ ,  but 3 is NOT related to 8 because no prime divide both numbers.