Question

In \(\mathbb{R}\), a relation \(p\) is defined as follows: For \(a, b \in \mathbb{R}\), \(apb\) holds if \(a^2 - 4ab + 3b^2 = 0\).
Then:

• A. p is an equivalence relation
• B. p is only symmetric
• C. p is only reflexiv
• D. p is only transitive

Hint:

To check if a relation is reflexive, test if apa holds for all elements a. If it does, the relation is reflexive.

Answer 1

The Correct Option is C

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.