For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, symmetric,antisymmetric, and it is transitive (d) {(1, 2), (2, 3), (3, 4)}

Let's analyze the relation \(\{(1, 2), (2, 3), (3, 4)\}\) 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 \(a \in A\), \((a, a) \in R\).

- 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 \(\{(1, 2), (2, 3), (3, 4)\}\), 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.

- Here, the relation includes \((1, 2)\), but not \((2, 1)\); it includes \((2, 3)\), but not \((3, 2)\); it includes \((3, 4)\), but not \((4, 3)\).
- Since none of the pairs have their corresponding reverse pairs in the relation, the relation is **not symmetric**.

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

- In this relation, there are no pairs \((a, b)\) and \((b, a)\) such that \(a \neq b\).
- Hence, the relation satisfies the condition trivially, so the relation is **antisymmetric**.

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

- We have \((1, 2) \in R\) and \((2, 3) \in R\), but \((1, 3) \notin R\).
- We also have \((2, 3) \in R\) and \((3, 4) \in R\), but \((2, 4) \notin R\).
- Since \((1, 3)\) and \((2, 4)\) are missing, the relation is **not transitive**.

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

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

Here are five questions that relate to this information:
1. What is the significance of antisymmetry in a relation?
2. How can a relation be both transitive and antisymmetric?
3. Why does the absence of certain pairs make a relation non-transitive?
4. What are some real-life examples of antisymmetric relations?
5. Can a relation be transitive without being reflexive?

**Tip:** To determine transitivity, carefully check for pairs that might be missing from the relation to ensure it satisfies the condition across all elements.

Explore the properties of relations on the set {1, 2, 3, 4} including reflexivity, symmetry, antisymmetry, and transitivity. Understand the definitions and implications of each property with a detailed analysis of the given relation {(1, 2), (2, 3), (3, 4)}.

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