etermine whether the each of the relation defined on the set of positive integers is reflexive,symmetric, antisymmetric, or transitive.(a) R = {(x, y) : xy = 2}(b) R = {(x, y) : xy ≥ 1}(c) R = {(x, y) : x = and2}(d) R = {(x, y) : 3 divides (x + 2and)}(It is) R = {(x, y) : x − and = 2}(f) R = {(x, y) : 3 divides (x − an

(a) R = {(x, y) : xy = 2}

• Reflexive: No, because for any positive integer x, x*x ≠ 2.
• Symmetric: Yes, because if (x, y) ∈ R, then xy = 2, and yx = 2, so (y, x) ∈ R.
• Antisymmetric: No, because if (x, y) ∈ R and (y, x) ∈ R, then x ≠ y.
• Transitive: No, because if (x, y) ∈ R and (y, z) ∈ R, then xy = 2 and yz = 2, but xz ≠ 2.

(b) R = {(x, y) : xy ≥ 1}

• Reflexive: Yes, because for any positive integer x, x*x ≥ 1.
• Symmetric: Yes, because if (x, y) ∈ R, then xy ≥ 1, and yx ≥ 1, so (y, x) ∈ R.
• Antisymmetric: No, because if (x, y) ∈ R and (y, x) ∈ R, x and y can be different.
• Transitive: Yes, because if (x, y) ∈ R and (y, z) ∈ R, then xy ≥ 1 and yz ≥ 1, so xz ≥ 1.

(c) R = {(x, y) : x = y}

• Reflexive: Yes, because for any positive integer x, x = x.
• Symmetric: Yes, because if (x, y) ∈ R, then x = y, and y = x, so (y, x) ∈ R.
• Antisymmetric: Yes, because if (x, y) ∈ R and (y, x) ∈ R, then x = y.
• Transitive: Yes, because if (x, y) ∈ R and (y, z) ∈ R, then x = y and y = z, so x = z.

(d) R = {(x, y) : 3 divides (x + y)}

• Reflexive: Yes, because for any positive integer x, 3 divides (x + x).
• Symmetric: Yes, because if (x, y) ∈ R, then 3 divides (x + y), and 3 divides (y + x), so (y, x) ∈ R.
• Antisymmetric: No, because if (x, y) ∈ R and (y, x) ∈ R, x and y can be different.
• Transitive: Yes, because if (x, y) ∈ R and (y, z) ∈ R, then 3 divides (x + y) and 3 divides (y + z), so 3 divides (x + z).

(e) R = {(x, y) : x − y = 2}

• Reflexive: No, because for any positive integer x, x - x ≠ 2.
• Symmetric: No, because if (x, y) ∈ R, then x - y = 2, but y - x ≠ 2.
• Antisymmetric: Yes, because if (x, y) ∈ R and (y, x) ∈ R, then x = y.
• Transitive: No, because if (x, y) ∈ R and (y, z) ∈ R, then x - y = 2 and y - z = 2, but x - z ≠ 2.

(f) R = {(x, y) : 3 divides (x - y)}

• Reflexive: Yes, because for any positive integer x, 3 divides (x - x).
• Symmetric: Yes, because if (x, y) ∈ R, then 3 divides (x - y), and 3 divides (y - x), so (y, x) ∈ R.
• Antisymmetric: No, because if (x, y) ∈ R and (y, x) ∈ R, x and y can be different.
• Transitive: Yes, because if (x, y) ∈ R and (y, z) ∈ R, then 3 divides (x - y) and 3 divides (y - z), so 3 divides (x - z).