Final answer:
The relation R is reflexive, symmetric, and transitive.
Step-by-step explanation:
To determine if the relation R is reflexive, we need to show that (p,q) R (p,q) for all (p,q) in Z. Since (p,q) R (r,s) if and only if (p+q) = (r+s), we can substitute (p,q) for both (r,s) and (p+q) for (r+s) in the relation to get (p+q) = (p+q), which is true for all (p,q) in Z. Therefore, R is reflexive.
To determine if the relation R is symmetric, we need to show that if (p,q) R (r,s), then (r,s) R (p,q). If (p,q) R (r,s), then (p+q) = (r+s). By rearranging the equation, we get (r+s) = (p+q), which implies (r,s) R (p,q). Therefore, R is symmetric.
To determine if the relation R is transitive, we need to show that if (p,q) R (r,s) and (r,s) R (u,v), then (p,q) R (u,v). If (p,q) R (r,s) and (r,s) R (u,v), then (p+q) = (r+s) and (r+s) = (u+v), respectively. By transitivity, we can substitute (p+q) for (u+v) and get (p,q) R (u,v). Therefore, R is transitive.