Final answer:
The relation P on the set of integers Z is not reflexive for all integers, but it is symmetric and transitive. This is determined by the properties of prime numbers and divisibility.
Step-by-step explanation:
Analysis of the Relation P Defined on the Integer Set Z
The question involves determining whether a relation P defined on the set of integers Z is reflexive, symmetric, and transitive. The relation is defined as follows: for any m, n in Z, mPn if and only if there exists a prime number p such that p divides m and p divides n. To analyze this, we need to understand the properties of the relation.
A relation P is reflexive if every element is related to itself. In this case, since every integer is divisible by at least one prime number (except for 1, which is not considered a prime, and 0, which is not divisible by any prime number as primes are positive), the relation is reflexive for all integers except 0 and 1. Consequently, the relation is not reflexive over the set of all integers Z.
A relation P is symmetric if for any elements a and b, if aPb, then bPa. This relation is symmetric because if a prime p divides both m and n, then it certainly divides both n and m.
A relation P is transitive if whenever aPb and bPc, then aPc also. This relation is transitive because if a prime p divides m (related to n) and the same or another prime divides n (related to o), then p or another common prime number will divide both m and o as well.
In summary, the given relation P is not reflexive over the entire set of integers Z, but it is symmetric and transitive when applicable.