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The relation R on the set A = {1, 2, 3} ...

The relation R on the set A = {1, 2, 3} defined as R ={(1, 1), (1, 2), (2, 1), (3, 3)} is reflexive, symmetric and transitive.

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To determine whether the relation \( R \) on the set \( A = \{1, 2, 3\} \) defined as \( R = \{(1, 1), (1, 2), (2, 1), (3, 3)\} \) is reflexive, symmetric, and transitive, we will check each property step by step. ### Step 1: Check for Reflexivity A relation \( R \) is reflexive if for every element \( a \) in set \( A \), the ordered pair \( (a, a) \) is in \( R \). - The elements in set \( A \) are \( 1, 2, 3 \). - We need to check for the pairs: - \( (1, 1) \) is in \( R \) (yes) - \( (2, 2) \) is not in \( R \) (no) - \( (3, 3) \) is in \( R \) (yes) Since \( (2, 2) \) is not in \( R \), the relation is **not reflexive**. ### Step 2: Check for Symmetry A relation \( R \) is symmetric if whenever \( (a, b) \) is in \( R \), then \( (b, a) \) is also in \( R \). - We will check each pair in \( R \): - For \( (1, 2) \), we check if \( (2, 1) \) is in \( R \) (yes, it is). - For \( (2, 1) \), we check if \( (1, 2) \) is in \( R \) (yes, it is). - For \( (1, 1) \), we check if \( (1, 1) \) is in \( R \) (yes, it is). - For \( (3, 3) \), we check if \( (3, 3) \) is in \( R \) (yes, it is). Since all pairs satisfy the symmetry condition, the relation is **symmetric**. ### Step 3: Check for Transitivity A relation \( R \) is transitive if whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \). - We will check each combination: - For \( (1, 2) \) and \( (2, 1) \): - We check if \( (1, 1) \) is in \( R \) (yes, it is). - For \( (2, 1) \) and \( (1, 2) \): - We check if \( (2, 2) \) is in \( R \) (no, it is not). - For \( (1, 1) \) and any other pair, it does not introduce new pairs. - For \( (3, 3) \), it does not introduce new pairs. Since we found a case where \( (2, 2) \) is not in \( R \) while \( (2, 1) \) and \( (1, 2) \) are in \( R \), the relation is **not transitive**. ### Conclusion - The relation \( R \) is **not reflexive**. - The relation \( R \) is **symmetric**. - The relation \( R \) is **not transitive**.
To determine whether the relation \( R \) on the set \( A = \{1, 2, 3\} \) defined as \( R = \{(1, 1), (1, 2), (2, 1), (3, 3)\} \) is reflexive, symmetric, and transitive, we will check each property step by step. ### Step 1: Check for Reflexivity A relation \( R \) is reflexive if for every element \( a \) in set \( A \), the ordered pair \( (a, a) \) is in \( R \). - The elements in set \( A \) are \( 1, 2, 3 \). - We need to check for the pairs: - \( (1, 1) \) is in \( R \) (yes) ...
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