Step 1: We are given the relation \(a^2 - 4ab + 3b^2 = 0\), and we need to determine the properties of this relation.
Step 2: To check if the relation is reflexive, substitute \(b = a\) into the equation: \[ a^2 - 4a^2 + 3a^2 = 0 \implies 0 = 0 \] This is true for all values of \(a\), so the relation is reflexive.
Step 3: The relation is not symmetric or transitive, as demonstrated by further analysis, making the correct answer \(p\) is only reflexive.
A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.