explain difference b/w symmetric, asymmetric and antisymmetric sets in simple language with simplest set example

Let's break down the concepts of symmetric, asymmetric, and antisymmetric sets with simple explanations and examples.

### 1. **Symmetric Relation:**
- **Definition:** A relation \( R \) on a set \( A \) is symmetric if, whenever an element \( a \) is related to an element \( b \) (i.e., \( aRb \)), then \( b \) is also related to \( a \) (i.e., \( bRa \)).
- **Example:** 
  - Set \( A = \{1, 2\} \)
  - Relation \( R = \{(1, 2), (2, 1)\} \)
  - Here, since \( (1, 2) \in R \) and \( (2, 1) \in R \), the relation is symmetric.

### 2. **Asymmetric Relation:**
- **Definition:** A relation \( R \) on a set \( A \) is asymmetric if, whenever an element \( a \) is related to an element \( b \) (i.e., \( aRb \)), then \( b \) is **not** related to \( a \) (i.e., \( bRa \) does **not** hold). Asymmetric relations are not symmetric at all.
- **Example:** 
  - Set \( A = \{1, 2\} \)
  - Relation \( R = \{(1, 2)\} \)
  - Here, \( (1, 2) \in R \) but \( (2, 1) \notin R \), so the relation is asymmetric.

### 3. **Antisymmetric Relation:**
- **Definition:** A relation \( R \) on a set \( A \) is antisymmetric if, whenever both \( aRb \) and \( bRa \) hold, then \( a \) must be equal to \( b \). This means if \( a \) is related to \( b \) and \( b \) is related to \( a \), they must be the same element.
- **Example:**
  - Set \( A = \{1, 2\} \)
  - Relation \( R = \{(1, 1), (2, 2), (1, 2)\} \)
  - Here, \( (1, 2) \in R \) but \( (2, 1) \notin R \). Also, \( (1, 1) \in R \) and \( (2, 2) \in R \), so the relation is antisymmetric.

### Summary:
- **Symmetric:** If \( a \) is related to \( b \), then \( b \) is related to \( a \).
- **Asymmetric:** If \( a \) is related to \( b \), then \( b \) is **not** related to \( a \).
- **Antisymmetric:** If \( a \) is related to \( b \) and \( b \) is related to \( a \), then \( a \) must be equal to \( b \).

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### Related Questions:
1. Can a relation be both symmetric and antisymmetric?
2. What is a real-life example of a symmetric relation?
3. How does transitivity relate to symmetric and antisymmetric relations?
4. What is a reflexive relation, and how does it differ from symmetric relations?
5. Can a relation be both asymmetric and antisymmetric?

### Tip:
When identifying the type of relation, always check pairs carefully to see if the condition holds for all elements in the set.

Learn about symmetric, asymmetric, and antisymmetric sets with clear explanations and examples. Understand how these concepts apply in set theory and relation definitions. Suitable for Grades 9-12 students and anyone interested in foundational set theory concepts.

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