Define Partial Order relation and check whether R is Partial Order relation.R= {(x,y) 𝑖𝑓 𝑦 = 𝑥𝑟, 𝑟 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑎𝑛𝑑 𝑎, 𝑏 ∈ 𝑍}

To define a partial order relation, we need to check three properties: reflexivity, antisymmetry, and transitivity.

1. Reflexivity: For every element x in the set, (x, x) must be in the relation R. In this case, if y = xr, where r is a positive integer and a, b are integers, then (x, y) = (x, xr) = (x, x^r) should be in R. Since x^r is always equal to x^r, this condition is satisfied.

2. Antisymmetry: For any two distinct elements x and y in the set, if (x, y) is in R, then (y, x) should not be in R. In this case, if y = xr and x = yr, where r is a positive integer and a, b are integers, then (x, y) = (xr, xr) = (x^r, x^r) is in R. However, (y, x) = (xr, xr) = (x^r, x^r) is also in R. Therefore, the antisymmetry property is not satisfied.

Since the antisymmetry property is not satisfied, the relation R is not a partial order relation.