How to determine the transitivity of a relation?

To determine the transitivity of a relation \( R \), you need to check if for any three elements \( a, b, \) and \( c \), whenever \( (a, b) \in R \) and \( (b, c) \in R \), it implies that \( (a, c) \in R \). This must hold true for all possible combinations of \( a, b, \) and \( c \) within the relation.

Here’s a step-by-step approach to verify the transitivity of a relation:

1. **List all pairs**: Write down all the pairs \((a, b)\) that belong to the relation \( R \).
2. **Check each pair for transitivity**: For each pair \((a, b)\), look for a pair \((b, c)\) in the relation.
3. **Verify the implication**: If you find such a pair \((b, c)\), then check if \((a, c)\) is also in the relation.
4. **Confirm for all pairs**: Repeat this for all pairs in the relation. If for every \((a, b)\) and \((b, c)\) pair, \((a, c)\) is in the relation, then \( R \) is transitive.

### Example:
Consider the relation \( R = \{(1, 2), (2, 3), (1, 3)\} \).

1. **List all pairs**: \( \{(1, 2), (2, 3), (1, 3)\} \)
2. **Check each pair**:
   - For \((1, 2)\), we find \((2, 3)\). We need to check if \((1, 3)\) is in the relation. It is.
   - For \((2, 3)\), there are no pairs starting with \( 3 \) in the relation, so we don't need to check further for this pair.
   - For \((1, 3)\), there are no pairs starting with \( 3 \) in the relation, so we don't need to check further for this pair.
3. **Verify the implication**: We already verified that \( (1, 3) \) is in the relation whenever \( (1, 2) \) and \( (2, 3) \) are in the relation.
4. **Confirm for all pairs**: Since we found no counterexample where \( (a, b) \in R \) and \( (b, c) \in R \) but \( (a, c) \notin R \), the relation is transitive.

Thus, the relation \( R \) is transitive.

### Summary
A relation \( R \) is transitive if:
\[ \forall a, b, c \in \text{set}, \ (a, b) \in R \text{ and } (b, c) \in R \Rightarrow (a, c) \in R. \]

Do you have any questions or need further clarification? Here are some related questions you might find interesting:

1. What are symmetric and reflexive relations?
2. How to check if a relation is symmetric?
3. What are equivalence relations?
4. How to represent relations using matrices?
5. What are some examples of non-transitive relations?

**Tip:** To systematically verify transitivity, you can use a matrix representation of the relation or draw a directed graph where an edge from node \(a\) to node \(b\) indicates \((a, b) \in R\).

Learn how to determine the transitivity of a relation with our detailed step-by-step guide. Understand the conditions under which a relation is transitive, along with examples and verification techniques. Suitable for advanced high school and college-level students.

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