If A = {1, 2, 3, 4}, define relations on A which have properties of being
(i) reflexive, transitive but not symmetric.
(ii) symmetric but neither reflexive nor transitive.
(iii) reflexive, symmetric and transitive.

Explanation

To solve the problem, we will define relations on the set \( A = \{1, 2, 3, 4\} \) that satisfy the specified properties for each part of the question.

### Part (i): Reflexive, Transitive but not Symmetric

1. **Reflexive Property**: A relation \( R \) is reflexive if every element in the set relates to itself. Therefore, we need to include the pairs \( (1,1), (2,2), (3,3), (4,4) \) in our relation \( R \).

So, we start with:
\[ R = \{(1,1), (2,2), (3,3), (4,4)\} \]

2. **Transitive Property**: A relation \( R \) is transitive if whenever \( (a,b) \in R \) and \( (b,c) \in R \), then \( (a,c) \in R \). To ensure transitivity, we can add pairs such as \( (1,2) \) and \( (2,1) \) to our relation. However, we need to ensure that the relation is not symmetric.

We can add:
\[ R = \{(1,1), (2,2), (3,3), (4,4), (1,2)\} \]
Here, \( (2,1) \) is not included, which ensures that the relation is not symmetric.

3. **Final Relation**:
\[ R = \{(1,1), (2,2), (3,3), (4,4), (1,2)\} \]

### Part (ii): Symmetric but neither Reflexive nor Transitive

1. **Symmetric Property**: A relation \( R \) is symmetric if whenever \( (a,b) \in R \), then \( (b,a) \in R \). We can choose pairs such as \( (1,2) \) and include \( (2,1) \) to satisfy symmetry.

We can start with:
\[ R = \{(1,2), (2,1)\} \]

2. **Not Reflexive**: To ensure that the relation is not reflexive, we will not include pairs like \( (1,1), (2,2), (3,3), (4,4) \).

3. **Not Transitive**: To ensure that the relation is not transitive, we can choose pairs that do not lead to a transitive closure. For example, if we add \( (3,4) \) and \( (4,3) \), we do not include \( (1,3) \) or \( (2,4) \).

Thus, we can have:
\[ R = \{(1,2), (2,1), (3,4), (4,3)\} \]

### Part (iii): Reflexive, Symmetric, and Transitive

1. **Reflexive Property**: We need to include all pairs where each element relates to itself:
\[ R = \{(1,1), (2,2), (3,3), (4,4)\} \]

2. **Symmetric Property**: We can add pairs such as \( (1,2) \) and \( (2,1) \) to ensure symmetry.

3. **Transitive Property**: To ensure transitivity, we can add pairs that connect the elements, such as \( (1,2), (2,3), (3,1) \), and their symmetric counterparts.

A possible relation that satisfies all three properties is:
\[ R = \{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1)\} \]

### Summary of Relations:
1. **Part (i)**: \( R = \{(1,1), (2,2), (3,3), (4,4), (1,2)\} \) (Reflexive, Transitive, Not Symmetric)
2. **Part (ii)**: \( R = \{(1,2), (2,1), (3,4), (4,3)\} \) (Symmetric, Not Reflexive, Not Transitive)
3. **Part (iii)**: \( R = \{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1)\} \) (Reflexive, Symmetric, Transitive)