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 need to define relations on the set \( A = \{1, 2, 3, 4\} \) that satisfy the given properties. Let's break it down step by step.

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

1. **Define the relation \( R \)**:
We can define the relation \( R \) as follows:
\[ R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3)\} \]

2. **Check if \( R \) is reflexive**:
A relation is reflexive if every element is related to itself. Here, we have:
- \( (1, 1) \)
- \( (2, 2) \)
- \( (3, 3) \)
- \( (4, 4) \)
All elements of \( A \) are included, so \( R \) is reflexive.

3. **Check if \( R \) is transitive**:
A relation is transitive if whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \).
- We have \( (1, 2) \) and \( (2, 3) \), so we need \( (1, 3) \) to be in \( R \) for transitivity.
- Adding \( (1, 3) \) to \( R \) gives us \( R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3), (1, 3)\} \).
- Now, we check:
- \( (1, 2) \) and \( (2, 3) \) implies \( (1, 3) \) is present.
- No other pairs violate transitivity.
Thus, \( R \) is transitive.

4. **Check if \( R \) is symmetric**:
A relation is symmetric if whenever \( (a, b) \) is in \( R \), then \( (b, a) \) must also be in \( R \).
- Here, \( (1, 2) \) is in \( R \), but \( (2, 1) \) is not in \( R \).
Therefore, \( R \) is not symmetric.

### Conclusion for (i):
The relation \( R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3), (1, 3)\} \) is reflexive and transitive but not symmetric.

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

1. **Define the relation \( R \)**:
We can define the relation \( R \) as follows:
\[ R = \{(1, 2), (2, 1)\} \]

2. **Check if \( R \) is symmetric**:
- Since \( (1, 2) \) is in \( R \) and \( (2, 1) \) is also in \( R \), the relation is symmetric.

3. **Check if \( R \) is reflexive**:
- For reflexivity, we need \( (1, 1) \), \( (2, 2) \), \( (3, 3) \), and \( (4, 4) \) to be in \( R \).
- None of these pairs are present, so \( R \) is not reflexive.

4. **Check if \( R \) is transitive**:
- For transitivity, we need that if \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \).
- Here, we have \( (1, 2) \) and \( (2, 1) \), but there are no pairs to satisfy transitivity.
- Since we can't find any \( (a, c) \) to satisfy the condition, we can say it is not transitive.

### Conclusion for (ii):
The relation \( R = \{(1, 2), (2, 1)\} \) is symmetric but neither reflexive nor transitive.

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

1. **Define the relation \( R \)**:
We can define the relation \( R \) as follows:
\[ R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)\} \]

2. **Check if \( R \) is reflexive**:
- All elements \( 1, 2, 3, 4 \) have their pairs \( (1, 1) \), \( (2, 2) \), \( (3, 3) \), and \( (4, 4) \) in \( R \).
Thus, \( R \) is reflexive.

3. **Check if \( R \) is symmetric**:
- For symmetry, we check:
- \( (1, 2) \) and \( (2, 1) \) are present.
- \( (2, 3) \) and \( (3, 2) \) are present.
- \( (1, 3) \) and \( (3, 1) \) are present.
Thus, \( R \) is symmetric.

4. **Check if \( R \) is transitive**:
- For transitivity:
- \( (1, 2) \) and \( (2, 3) \) imply \( (1, 3) \) is present.
- \( (2, 1) \) and \( (1, 3) \) imply \( (2, 3) \) is present.
- All combinations satisfy transitivity.
Thus, \( R \) is transitive.

### Conclusion for (iii):
The relation \( R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)\} \) is reflexive, symmetric, and transitive.

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