can i conclude that antisymmetric is combination of reflexive and asymmetric sets

No, you cannot conclude that antisymmetric is simply a combination of reflexive and asymmetric properties. Let’s clarify the relationship between these concepts:

### **Key Points:**
1. **Reflexive Relation:**
   - A relation \( R \) on a set \( A \) is reflexive if every element is related to itself. That is, for every \( a \in A \), \( (a, a) \in R \).
   - Example: For set \( A = \{1, 2\} \), if \( R = \{(1, 1), (2, 2)\} \), then \( R \) is reflexive.

2. **Asymmetric Relation:**
   - A relation \( R \) is asymmetric if whenever \( (a, b) \in R \), \( (b, a) \) is **not** in \( R \).
   - Example: If \( R = \{(1, 2)\} \), then \( R \) is asymmetric because \( (2, 1) \notin R \).

3. **Antisymmetric Relation:**
   - A relation \( R \) is antisymmetric if whenever both \( (a, b) \) and \( (b, a) \) are in \( R \), then \( a \) must be equal to \( b \). This allows for reflexive pairs like \( (a, a) \).
   - Example: If \( R = \{(1, 1), (2, 2), (1, 2)\} \), then \( R \) is antisymmetric because there are no pairs \( (a, b) \) and \( (b, a) \) where \( a \neq b \).

### **Differences:**
- **Antisymmetric ≠ Asymmetric + Reflexive:**
   - **Asymmetric** relations prohibit any pair \( (a, b) \) from having \( (b, a) \) in the relation, regardless of whether \( a = b \).
   - **Antisymmetric** relations allow pairs like \( (a, a) \) (which is a reflexive property) but require that if \( (a, b) \) and \( (b, a) \) exist, then \( a \) must equal \( b \).

### **Example to Clarify:**
- Consider the set \( A = \{1, 2\} \).
- **Reflexive but not Asymmetric:** 
   - \( R = \{(1, 1), (2, 2)\} \) is reflexive, but not asymmetric because no \( (a, b) \) pair exists where \( a \neq b \).
- **Antisymmetric and Reflexive:**
   - \( R = \{(1, 1), (2, 2)\} \) is both reflexive and antisymmetric, but it's not asymmetric because asymmetric relations wouldn’t include \( (1, 1) \) or \( (2, 2) \) if it were asymmetric.

In conclusion, **antisymmetric** is a distinct concept and cannot be simply viewed as a combination of **reflexive** and **asymmetric** relations. They each have unique definitions and conditions.

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Would you like to explore this further, or do you have any questions?

### Related Questions:
1. How do antisymmetric relations behave on larger sets?
2. What is the difference between symmetric and antisymmetric relations in terms of examples?
3. Can a relation be neither symmetric, asymmetric, nor antisymmetric?
4. What are examples of antisymmetric relations in real-world applications?
5. How does a transitive relation interact with antisymmetry?

### Tip:
When dealing with relations, using simple examples can help clarify the sometimes subtle differences between asymmetric and antisymmetric properties.

Explore the definitions and differences between reflexive, asymmetric, and antisymmetric relations in mathematics. Learn with examples and clarify how each concept operates within different sets. Suitable for undergraduate level studies.

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