For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, symmetric, 

antisymmetric, and it is transitive.

(a) {(2, 4), (4, 2)}

Let's analyze the relation \(\{(2, 4), (4, 2)\}\) on the set \(\{1, 2, 3, 4\}\) with respect to the properties of reflexivity, symmetry, antisymmetry, and transitivity.

### 1. Reflexivity
A relation \(R\) on a set \(A\) is **reflexive** if every element is related to itself. That is, for every \(a \in A\), \((a, a) \in R\).

- Here, the set is \(\{1, 2, 3, 4\}\). 
- For the relation to be reflexive, it must include \((1, 1)\), \((2, 2)\), \((3, 3)\), and \((4, 4)\).
- Since none of these pairs are in the relation \(\{(2, 4), (4, 2)\}\), the relation is **not reflexive**.

### 2. Symmetry
A relation \(R\) is **symmetric** if for every \((a, b) \in R\), \((b, a) \in R\) as well.

- In this relation, \((2, 4)\) is present, and \((4, 2)\) is also present.
- Since both pairs \((2, 4)\) and \((4, 2)\) are present, the relation is **symmetric**.

### 3. Antisymmetry
A relation \(R\) is **antisymmetric** if for every \((a, b) \in R\) and \((b, a) \in R\), it must be that \(a = b\).

- Here, both \((2, 4)\) and \((4, 2)\) are in the relation, but \(2 \neq 4\).
- Thus, the relation is **not antisymmetric**.

### 4. Transitivity
A relation \(R\) is **transitive** if whenever \((a, b) \in R\) and \((b, c) \in R\), then \((a, c) \in R\).

- In this relation, there are no such pairs \((a, b)\) and \((b, c)\) where \(b\) matches the second element of the first pair and the first element of the second pair. 
- Thus, there is no violation of transitivity, and the relation is **transitive**.

### Summary
- **Reflexive**: No
- **Symmetric**: Yes
- **Antisymmetric**: No
- **Transitive**: Yes

Would you like more details on any of these properties or further examples?

Here are five questions that relate to this information:
1. What makes a relation reflexive?
2. Can a relation be both symmetric and antisymmetric?
3. How do you determine if a relation is transitive?
4. What happens if a relation is only partially reflexive?
5. How do these properties apply to larger sets?

**Tip:** Always check the pairs in a relation methodically for each property to avoid confusion, especially in complex relations.

Explore the analysis of a relation on the set {1, 2, 3, 4} concerning reflexivity, symmetry, antisymmetry, and transitivity. Understand key concepts in relation theory with detailed explanations and examples.

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