Question

Let R be the relation "has-the-same-size-as" defined on all finite subsets of Z. Which of the five properties (reflexive, irreflexive, symmetric, antisymmetric, transitive) does R have? Prove your answers. Yes No • Reflexive? Circle one: Explanation: Yes No • Irreflexive? Circle one: Explanation: Yes No • Symmetric? Circle one: Explanation: Yes No • Antisymmetric? Circle one: Explanation: Yes No • Transitive? Circle one: Explanation

Answer 1

- Reflexive? No

- Irreflexive? Yes

- Symmetric? Yes

- Antisymmetric? No

- Transitive? Yes

Step-by-step explanation:

The relation "has-the-same-size-as" (R) on finite subsets of Z (integers) is not reflexive because a set isn't the same size as itself unless it's an empty set. It is irreflexive because no non-empty set can have the same size as itself.

Symmetry holds since if set A has the same size as set B, then B also has the same size as A. For instance, if set A has 3 elements and set B has 3 elements, they're of equal size.

However, R isn't antisymmetric because even though if A has the same size as B and vice versa, it doesn't mean that A and B have to be identical. For instance, if set A has 2 elements and set B also has 2 elements but they aren't the same set, it breaks the antisymmetric property.

Regarding transitivity, if A has the same size as B and B has the same size as C, then A has the same size as C. For instance, if A has 4 elements, B has 4 elements, and C has 4 elements, they're all of the same size, fulfilling the transitive property.

Answer 2

Reflexive:(No) The empty set doesn't have the same size as itself.

Irreflexive: (Yes) No finite subset can have the same size as itself (excluding the empty set).

Symmetric: (Yes) If A has the same size as B, then B has the same size as A.

Antisymmetric: (No) Having the same size doesn't guarantee equal sets.

Transitive: (Yes) If A has the same size as B, and B has the same size as C, then A has the same size as C.

Properties of the relation "has-the-same-size-as" on finite subsets of Z:

1. Reflexive? No

Consider the empty set, {}. It does not have the same size as itself, so R is not reflexive for all elements in Z.

2. Irreflexive? Yes

No finite subset of Z has the same size as itself (excluding the empty set, which we already considered). Therefore, R is irreflexive for all non-empty elements in Z.

3. Symmetric? Yes

If a set A has the same size as a set B (A R B), then B also has the same size as A (B R A). The size relationship is symmetrical.

4. Antisymmetric? No

If A R B and B R A, it doesn't necessarily mean A = B. For example, {1, 2} and {2, 1} have the same size (both are 2), but they are not the same sets.

5. Transitive? Yes

If A R B (A has the same size as B) and B R C (B has the same size as C), then A R C (A has the same size as C). The size relationship is transitive.