Given the S = {p, q, r, s, t}, determine if the following relations R on Sis symmetric, asymmetric, and (or) antisymmetric?
(i)R = {(q,q), (q,r)}
(ii)R = {(p,p), (s,s), (t,t)}
(iii)R = {(r,r), (s,s), (s,t) (t,s)}}
(iv)R = {(q,q), (s,s)}

Let's analyze each relation \( R \) on the set \( S = \{p, q, r, s, t\} \) to determine if they are symmetric, asymmetric, and/or antisymmetric.

### Key definitions:
- **Symmetric**: A relation \( R \) is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \) as well.
- **Asymmetric**: A relation \( R \) is asymmetric if whenever \( (a, b) \in R \), then \( (b, a) \notin R \) for all \( a \) and \( b \).
- **Antisymmetric**: A relation \( R \) is antisymmetric if whenever \( (a, b) \in R \) and \( (b, a) \in R \), then \( a = b \).

### (i) \( R = \{(q, q), (q, r)\} \)

- **Symmetric**: To check symmetry, we look at \( (q, r) \in R \). Since \( (r, q) \notin R \), the relation is **not symmetric**.
- **Asymmetric**: Since \( (q, q) \in R \) (a self-loop), the relation cannot be asymmetric, because asymmetry requires no \( (a, a) \in R \). Hence, it is **not asymmetric**.
- **Antisymmetric**: Antisymmetry holds because there is no pair \( (q, r) \) and \( (r, q) \) in \( R \), and self-loops like \( (q, q) \) do not violate antisymmetry. Therefore, the relation is **antisymmetric**.

### (ii) \( R = \{(p, p), (s, s), (t, t)\} \)

- **Symmetric**: This relation consists entirely of self-loops, which are trivially symmetric. Thus, the relation is **symmetric**.
- **Asymmetric**: The presence of self-loops (e.g., \( (p, p) \)) makes this relation **not asymmetric**.
- **Antisymmetric**: Since there are no pairs \( (a, b) \) and \( (b, a) \) for distinct \( a \neq b \), the relation is **antisymmetric**.

### (iii) \( R = \{(r, r), (s, s), (s, t), (t, s)\} \)

- **Symmetric**: Here, \( (s, t) \in R \) and \( (t, s) \in R \), so the relation is **symmetric**.
- **Asymmetric**: Since \( (s, t) \in R \) and \( (t, s) \in R \), the relation is **not asymmetric**.
- **Antisymmetric**: Antisymmetry requires that if \( (a, b) \in R \) and \( (b, a) \in R \), then \( a = b \). Here, \( (s, t) \in R \) and \( (t, s) \in R \), but \( s \neq t \), so the relation is **not antisymmetric**.

### (iv) \( R = \{(q, q), (s, s)\} \)

- **Symmetric**: This relation consists entirely of self-loops, so it is **symmetric**.
- **Asymmetric**: The presence of self-loops makes the relation **not asymmetric**.
- **Antisymmetric**: Since there are no pairs \( (a, b) \) and \( (b, a) \) for distinct \( a \neq b \), the relation is **antisymmetric**.

### Summary of properties:

| Relation \( R \)   | Symmetric | Asymmetric | Antisymmetric |
|--------------------|-----------|------------|---------------|
| \( (i) \)          | No        | No         | Yes           |
| \( (ii) \)         | Yes       | No         | Yes           |
| \( (iii) \)        | Yes       | No         | No            |
| \( (iv) \)         | Yes       | No         | Yes           |

Would you like more details on any specific concept or additional questions?

### Further related questions:
1. Can a relation be both symmetric and antisymmetric at the same time?
2. How does a relation on a set change if we add or remove pairs?
3. What are some real-world examples of symmetric and asymmetric relations?
4. Is it possible for a relation to be asymmetric and antisymmetric at the same time?
5. How would the properties change if we modified the set \( S \)?

### Tip:
A relation with only self-loops is both symmetric and antisymmetric but never asymmetric.

Given the S = {p, q, r, s, t}, determine if the following relations R on S is symmetric, asymmetric, and (or) antisymmetric?
(i) R = {(q,q), (q,r)}
(ii) R = {(p,p), (s,s), (t,t)}
(iii) R = {(r,r), (s,s), (s,t), (t,s)}
(iv) R = {(q,q), (s,s)}

Analyze the relations R on the set S = {p, q, r, s, t} to determine if they are symmetric, asymmetric, and/or antisymmetric. Explore key concepts in set theory, including symmetry, antisymmetry, and asymmetry with detailed explanations. Suitable for students in Grades 10-12.

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