Question

Let R be a binary relation on the set of all positive integers given by (x, y) ∈ R if and only if xy ≥ 10. Determine if this relation reflexive, symmetric, transitive, or antisymmetric. Prove your answer.

Answer 1

The given relation is not reflexive, not symmetric, transitive, and not antisymmetric.

Step-by-step explanation:

The given binary relation R on the set of positive integers is defined as (x, y) ∈ R if and only if xy ≥ 10. To determine the properties of this relation, let's analyze each property:

1. Reflexive: A relation is reflexive if for every element x in the set, (x, x) is in the relation. In this case, when x = y = 1, we have 1*1 = 1, which is not greater than or equal to 10. Therefore, (1, 1) is not in R, and the relation is not reflexive.
2. Symmetric: A relation is symmetric if whenever (x, y) is in the relation, then (y, x) is also in the relation. Here, if (x, y) is in R, it means that xy ≥ 10. But xy is not necessarily equal to yx, so the relation is not symmetric.
3. Transitive: A relation is transitive if whenever (x, y) and (y, z) are in the relation, then (x, z) is also in the relation. If xy ≥ 10 and yz ≥ 10, then xz = (xy)(yz) ≥ 10*10 = 100, which means (x, z) is in R. Therefore, the relation is transitive.
4. Antisymmetric: A relation is antisymmetric if for any distinct elements x and y, if (x, y) is in the relation, then (y, x) is not in the relation. In this case, if xy ≥ 10 and yx ≥ 10, it means that xy and yx are both greater than or equal to 10. But it is not necessary that xy = yx, so the relation is not antisymmetric.

In conclusion, the given relation is not reflexive, not symmetric, transitive, and not antisymmetric.

Answer 2

The binary relation R is symmetric but not reflexive, not definitively transitive, and not antisymmetric, as it fails to meet the necessary conditions for these properties with the exception of symmetry.

Step-by-step explanation:

To determine if the given binary relation R where (x, y) ∈ R if and only if xy ≥ 10 is reflexive, symmetric, transitive, or antisymmetric, we need to evaluate it based on the definitions of those properties:

• Reflexive: A relation R is reflexive if every element is related to itself. For (x, x) to be in R, we would need xx ≥ 10, which is true for all x ≥ 4. However, for x = 1, 2, or 3, it is not true that xx ≥ 10, so R is not reflexive.
• Symmetric: A relation is symmetric if for every (x, y) in R, we also have (y, x) in R. Given that xy ≥ 10, we can also say yx ≥ 10 (since multiplication is commutative), so R is symmetric.
• Transitive: A relation is transitive if whenever (x, y) and (y, z) are in R, then (x, z) is also in R. If xy ≥ 10 and yz ≥ 10, then xz could be smaller or equal to 10 if y is less than 10. Thus, we cannot definitely say the relation is transitive without further information about the values of x, y, and z.
• Antisymmetric: A relation is antisymmetric if for every (x, y) and (y, x) in R where x ≠ y, it holds that R contains only one of these two elements. Since R is already established as symmetric, it cannot be antisymmetric.

In conclusion, the relation R on the set of all positive integers where (x, y) ∈ R if and only if xy ≥ 10 is symmetric but not reflexive, not definitively transitive, and not antisymmetric.