Define a transitive relation.

A transitive relation is a fundamental concept in mathematics, particularly in set theory and relational algebra. Here's the detailed explanation in plain text:

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**Definition:**

A relation \( R \) on a set \( A \) is called *transitive* if, whenever \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is also related to \( c \). Formally:

For all \( a, b, c \in A \), if \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).

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**Explanation:**

1. A *relation* on a set \( A \) is a subset of the Cartesian product \( A \times A \), i.e., a collection of ordered pairs \( (a, b) \) where \( a, b \in A \).
2. A relation is transitive if it satisfies the property described above.

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**Example 1 (Transitive Relation):**
Let \( A = \{1, 2, 3\} \) and \( R = \{(1, 2), (2, 3), (1, 3)\} \).
- \( (1, 2) \in R \) and \( (2, 3) \in R \) imply \( (1, 3) \in R \).
- Since this condition holds for all pairs in \( R \), \( R \) is transitive.

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**Example 2 (Non-Transitive Relation):**
Let \( A = \{1, 2, 3\} \) and \( R = \{(1, 2), (2, 3)\} \).
- \( (1, 2) \in R \) and \( (2, 3) \in R \), but \( (1, 3) \notin R \).
- Since the condition fails, \( R \) is not transitive.

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**Key Points:**
- Transitivity is a property of the relation, not the set.
- A transitive relation may or may not include all possible ordered pairs in \( A \times A \); it only needs to satisfy the transitive property.