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Give an example of a relation which ...

Give an example of a relation which is
(i) Reflexive and transitive but not symmetric,
(ii) symmetric and transitive but not Reflexive ,
(iii) reflexive and symmetric not transitive,
(iv) symmetric but neither reflexive nor transitive,
(v) transitive but neither reflexive nor symmetric.

Text Solution

AI Generated Solution

To solve the problem, we need to provide examples of relations that satisfy specific properties: reflexive, symmetric, and transitive. Let's break down each part step by step. ### (i) Reflexive and Transitive but Not Symmetric 1. **Define the Set**: Let \( A = \{1, 2, 3\} \). 2. **Define the Relation**: Let \( R_1 = \{(1, 1), (2, 2), (3, 3), (1, 2)\} \). 3. **Check Reflexivity**: A relation is reflexive if for every element \( x \in A \), the pair \( (x, x) \) is in \( R \). Here, \( (1, 1), (2, 2), (3, 3) \) are present, so \( R_1 \) is reflexive. 4. **Check Transitivity**: A relation is transitive if whenever \( (x, y) \) and \( (y, z) \) are in \( R \), then \( (x, z) \) must also be in \( R \). Here, \( (1, 1) \) and \( (1, 2) \) implies \( (1, 2) \) is present. Thus, \( R_1 \) is transitive. 5. **Check Symmetry**: A relation is symmetric if whenever \( (x, y) \) is in \( R \), then \( (y, x) \) must also be in \( R \). Here, \( (1, 2) \) is present but \( (2, 1) \) is not, so \( R_1 \) is not symmetric. ...
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