https://file.aiquickdraw.com/mathbot/file/17290003908206mlq28a6.jpg

The problem asks for the smallest relation that contains the given set \(\{(1,1), (1,2), (2,2), (3,1)\}\) and is both transitive and reflexive. It hints that the key is the "closure."

### Steps to solve:

1. **Reflexive Closure**: 
   A reflexive relation must contain \((a, a)\) for all elements \(a\) in the set. Looking at the elements in the given pairs, the set of elements is \(\{1, 2, 3\}\). To ensure reflexivity, we need to add:
   - \((3,3)\), since we already have \((1,1)\) and \((2,2)\).

2. **Transitive Closure**: 
   A transitive relation must satisfy: if \((a,b)\) and \((b,c)\) are in the relation, then \((a,c)\) must also be included. Now, let's analyze transitivity:
   - We have \((1,2)\) and \((2,2)\), so we must add \((1,2)\) again (but it's already present).
   - We have \((3,1)\) and \((1,2)\), so we must add \((3,2)\).

3. **Final Relation**:
   The relation now contains:
   - Reflexive closure: \(\{(1,1), (2,2), (3,3)\}\)
   - Original pairs: \(\{(1,2), (3,1)\}\)
   - Additional pairs for transitivity: \(\{(3,2)\}\)

Thus, the smallest relation that contains \(\{(1,1), (1,2), (2,2), (3,1)\}\) and is both transitive and reflexive is:
\[
\{(1,1), (1,2), (2,2), (3,1), (3,2), (3,3)\}
\]

Would you like more details on any of these steps?

### Related Questions:
1. What are the definitions of reflexive and transitive relations?
2. How do you compute the transitive closure of a relation?
3. Can a relation be transitive but not reflexive?
4. What is the significance of reflexive closure in relations?
5. How does the transitive property affect the structure of a relation?

### Tip:
When working with relations, it’s helpful to first break the relation down by closure properties (reflexive, symmetric, transitive) to simplify the process.

Carilah relasi terkecil yang mengandung relasi {(1,1), (1,2), (2,2), (3,1)} dan bersifat menghantar dan refleksif sekaligus.

Learn how to find the smallest relation containing the set {(1,1), (1,2), (2,2), (3,1)} that is both transitive and reflexive. This problem explores key concepts like reflexive and transitive closure in relations, offering a detailed step-by-step solution. Suitable for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation