Question

The relation \( R = \{(a, a), (b, b), (c, c), (a, c)\ \) defined on the set \( \{a, b, c\} \) is _____________.}

Hint:

Always check the properties of reflexivity, symmetry, and transitivity to classify relations on sets. A relation can be reflexive but still fail to be symmetric or transitive.

Answer 1

Step 1: To determine the type of relation, we need to analyze the properties of the given relation \( R \) on the set \( \{a, b, c\} \). The relation \( R \) is defined as: \[ R = \{(a, a), (b, b), (c, c), (a, c)\}. \] 

Step 2: Check for the following properties: 
- Reflexive: A relation is reflexive if for every element \( x \in S \), \( (x, x) \) belongs to the relation. In this case, \( (a, a), (b, b), (c, c) \) are in the relation, so the relation is reflexive. 
- Symmetric: A relation is symmetric if for every pair \( (x, y) \) in the relation, the pair \( (y, x) \) is also in the relation. Since \( (a, c) \) is in the relation but \( (c, a) \) is not, the relation is not symmetric. 
- Transitive: A relation is transitive if whenever \( (x, y) \) and \( (y, z) \) are in the relation, \( (x, z) \) must also be in the relation. Since \( (a, c) \) is in the relation but there is no corresponding \( (c, a) \), the relation is not transitive. 

Step 3: The relation is reflexive but not symmetric or transitive, which makes it spontaneous, traditional, but not conformist. Thus, the relation \( R \) is spontaneous, traditional, but not conformist.