Given the relation **R={(2, 4), (4, 2)}** on the set **A={1, 2, 3, 4}**. 

Which of the following is correct?

**Points will be deducted for each incorrect response; however, negative scoring will not be applied.**

Question 2 Select one or more:

Irreflexive, Not antisymmetric and Not transitive


Not reflexive, Not antisymmetric and transitive

Irreflexive, antisymmetric and Not transitive


Reflexive, Not antisymmetric and symmetric


Not reflexive, antisymmetric and transitive


Irreflexive, antisymmetric and transitive


None of the listed answers is correct.


Not reflexive, Not antisymmetric and Not transitive


Reflexive, Not antisymmetric and Not symmetric


Not reflexive, antisymmetric and symmetric   


Not reflexive, antisymmetric and Not transitive


Reflexive, antisymmetric and Not symmetric


Irreflexive, Not antisymmetric and transitive

To analyze the relation \( R = \{(2, 4), (4, 2)\} \) on the set \( A = \{1, 2, 3, 4\} \), let's examine its properties: reflexivity, irreflexivity, antisymmetry, symmetry, and transitivity.

### 1. **Reflexivity:**
A relation is reflexive if for every element \( a \in A \), the pair \( (a, a) \) is in \( R \). 
- Since \( R \) does not contain any pairs like \( (1, 1), (2, 2), (3, 3), \) or \( (4, 4) \), the relation is **not reflexive**.

### 2. **Irreflexivity:**
A relation is irreflexive if for every element \( a \in A \), the pair \( (a, a) \) is **not** in \( R \). 
- As no pair of the form \( (a, a) \) exists in \( R \), the relation is **irreflexive**.

### 3. **Antisymmetry:**
A relation is antisymmetric if for all \( a, b \in A \), whenever \( (a, b) \in R \) and \( (b, a) \in R \), it must be that \( a = b \).
- Here, \( (2, 4) \) and \( (4, 2) \) are both in \( R \), but \( 2 \neq 4 \). Therefore, the relation is **not antisymmetric**.

### 4. **Symmetry:**
A relation is symmetric if for every \( (a, b) \in R \), \( (b, a) \) is also in \( R \).
- Since \( (2, 4) \) and \( (4, 2) \) are both in \( R \), the relation is **symmetric**.

### 5. **Transitivity:**
A relation is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \) must also be in \( R \).
- For \( (2, 4) \) and \( (4, 2) \), there is no \( (2, 2) \) in \( R \), nor does any other pair satisfy this condition. Thus, the relation is **not transitive**.

### Conclusion:
Based on the analysis:
- The relation is **irreflexive** (no \( (a, a) \) pairs in \( R \)).
- The relation is **not antisymmetric** (since \( (2, 4) \) and \( (4, 2) \) are in \( R \)).
- The relation is **not transitive**.
- The relation is **symmetric**.

The correct options are:
- **Irreflexive, Not antisymmetric and Not transitive**
- **Not reflexive, Not antisymmetric and Not transitive**

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Would you like further details or have any questions? Here are some related questions:

1. How is antisymmetry different from symmetry in relations?
2. What are some examples of irreflexive relations in everyday life?
3. How can we determine if a relation is an equivalence relation?
4. What are some practical applications of transitive relations?
5. Can a relation be both symmetric and antisymmetric? Why or why not?
6. How does reflexivity relate to the concept of identity in mathematics?
7. What happens if a relation is both reflexive and transitive?
8. Can a non-reflexive relation be transitive?

**Tip:** When determining the properties of a relation, systematically check each property (reflexivity, symmetry, transitivity, etc.) to ensure a complete understanding.

Understand the properties of relations through a detailed analysis of reflexivity, irreflexivity, antisymmetry, symmetry, and transitivity. Learn how to determine these properties using examples and explanations. Suitable for college-level students and those interested in discrete mathematics.

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