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Let A = {1,2,3} and consider the relatio...

Let A = {1,2,3} and consider the relation R = (1,1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3),(3,1)}. Then R is

A

reflexive but not symmetric

B

reflexive but not transitive

C

symmetric and transitive

D

neither symmetric, nor transitive

Text Solution

AI Generated Solution

The correct Answer is:
To determine the properties of the relation \( R \) defined on the set \( A = \{1, 2, 3\} \), we need to check if \( R \) is reflexive, symmetric, and transitive. Given the relation \( R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3), (3,1)\} \), we will check each property step by step. ### Step 1: Check for Reflexivity A relation \( R \) is reflexive if every element \( a \) in set \( A \) is related to itself. This means that for every \( a \in A \), the pair \( (a, a) \) must be in \( R \). - For \( 1 \): \( (1,1) \in R \) (present) - For \( 2 \): \( (2,2) \in R \) (present) - For \( 3 \): \( (3,3) \in R \) (present) Since all pairs \( (1,1), (2,2), (3,3) \) are present in \( R \), the relation \( R \) is **reflexive**. ### Step 2: Check for Symmetry A relation \( R \) is symmetric if for every pair \( (a, b) \in R \), the pair \( (b, a) \) must also be in \( R \). - Check \( (1,2) \): \( (2,1) \notin R \) (missing) - Check \( (2,3) \): \( (3,2) \notin R \) (missing) - Check \( (1,3) \): \( (3,1) \in R \) (present) Since \( (2,1) \) and \( (3,2) \) are not in \( R \), the relation \( R \) is **not symmetric**. ### Step 3: Check for Transitivity A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \) must also be in \( R \). - Check \( (1,2) \) and \( (2,3) \): \( (1,3) \in R \) (present) - Check \( (3,1) \) and \( (1,2) \): \( (3,2) \notin R \) (missing) - Check \( (1,3) \) and \( (3,1) \): \( (1,1) \in R \) (present) Since \( (3,2) \) is not in \( R \) while \( (3,1) \) and \( (1,2) \) are in \( R \), the relation \( R \) is **not transitive**. ### Conclusion The relation \( R \) is: - Reflexive: Yes - Symmetric: No - Transitive: No Thus, the final answer is that \( R \) is **reflexive but not symmetric and not transitive**.
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