1.       Explain 4 properties of relation with example.

### Properties of Relations in Mathematics

A **relation** in mathematics is a connection between elements of two sets. Its properties are fundamental in set theory and relational algebra. Below are **four key properties of relations** with examples:

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### 1. **Reflexive Property**
A relation \( R \) on a set \( A \) is **reflexive** if every element is related to itself.

#### Definition:
\[
\forall a \in A, \ (a, a) \in R
\]

#### Example:
Let \( A = \{1, 2, 3\} \), and \( R = \{(1,1), (2,2), (3,3), (1,2)\} \).  
This relation \( R \) is **reflexive** because every element in \( A \) appears as a pair \((a, a)\) in \( R \).

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### 2. **Symmetric Property**
A relation \( R \) on a set \( A \) is **symmetric** if whenever \((a, b) \in R\), then \((b, a) \in R\).

#### Definition:
\[
\forall a, b \in A, \ (a, b) \in R \implies (b, a) \in R
\]

#### Example:
Let \( A = \{1, 2, 3\} \), and \( R = \{(1,2), (2,1), (2,3), (3,2)\} \).  
This relation \( R \) is **symmetric** because for every \((a, b)\), the pair \((b, a)\) is also present.

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### 3. **Transitive Property**
A relation \( R \) on a set \( A \) is **transitive** if whenever \((a, b) \in R\) and \((b, c) \in R\), then \((a, c) \in R\).

#### Definition:
\[
\forall a, b, c \in A, \ (a, b) \in R \ \text{and} \ (b, c) \in R \implies (a, c) \in R
\]

#### Example:
Let \( A = \{1, 2, 3\} \), and \( R = \{(1,2), (2,3), (1,3)\} \).  
This relation \( R \) is **transitive** because \((1,2)\) and \((2,3)\) imply \((1,3)\).

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### 4. **Antisymmetric Property**
A relation \( R \) on a set \( A \) is **antisymmetric** if whenever \((a, b) \in R\) and \((b, a) \in R\), then \(a = b\).

#### Definition:
\[
\forall a, b \in A, \ (a, b) \in R \ \text{and} \ (b, a) \in R \implies a = b
\]

#### Example:
Let \( A = \{1, 2, 3\} \), and \( R = \{(1,1), (2,2), (3,3), (1,2)\} \).  
This relation \( R \) is **antisymmetric** because there are no pairs where \((a, b)\) and \((b, a)\) both exist unless \(a = b\).

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### Recap Table of Examples:
| Property       | Key Condition                    | Example Relation \( R \)       | Valid?  |
|----------------|----------------------------------|---------------------------------|---------|
| Reflexive      | \((a, a) \in R\) for all \(a\)   | \( \{(1,1), (2,2), (3,3)\} \)  | Yes     |
| Symmetric      | \((a, b) \in R \implies (b, a) \in R\) | \( \{(1,2), (2,1)\} \)        | Yes     |
| Transitive     | \((a, b), (b, c) \in R \implies (a, c) \in R\) | \( \{(1,2), (2,3), (1,3)\} \) | Yes     |
| Antisymmetric  | \((a, b), (b, a) \in R \implies a = b\) | \( \{(1,2), (2,1)\} \)        | No      |

Would you like further examples or detailed proofs for these properties?  

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### Relative Questions:
1. How do reflexive, symmetric, and transitive properties relate to equivalence relations?
2. What is the difference between symmetric and antisymmetric relations?
3. Can a relation be both symmetric and antisymmetric? Explain with an example.
4. What are some real-world examples of reflexive relations?
5. How do these properties of relations apply to graphs and digraphs?

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### Tip:
When analyzing a relation's properties, always verify them with specific pairs in the set to avoid overlooking exceptions!

Explain 4 properties of relation with example.

Explore the four key properties of relations in set theory: reflexive, symmetric, transitive, and antisymmetric, each explained with examples. Ideal for high school students studying relational algebra and set theory.

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