Question

If a relation $R$ on the set $N$of natural numbers is defined as$(x,y)\iff x^2-4xy+3y^2=0,x,y$ is an element of $N$. Then, the relation $R$ is

• A. reflexive
• B. symmetric
• C. transitive
• D. an equivalence relation

Answer 1

The correct option is A

reflexive

Explanation for the correct option:

Finding the relation R:

Given relation is $(x,y)\iff x^2-4xy+3y^2=0,x,y\in N$

$$\begin{gathered} x^2–4xy+3y^2=0 \\ \Rightarrow (x–y)(x–3y)=0 \end{gathered}$$

Let $(a,a)\in R$

$\Rightarrow (a–a)(a–3a)=0$

Relation is reflexive.

Hence, the correct option is A.

The explanation of the incorrect option :

Option B: Symmetric :

For symmetric $(a,b)\in R\Rightarrow (b,a)\in R$

$(3,1)\in R\because (3–1)(3–3)=0$

But $(1,3)\notin R\because (1–3)(1–9)=16\ne 0$

It is not symmetric.

Option B: Transitive :

For transitive $(a,b)\in R;(b,c)\in R\Rightarrow (a,c)\in R$

But $(9,3)\in R;(3,1)\in R\Rightarrow (9,1)\notin R$

It is not transitive.

Option D : Symmetric :

Therefore it is not an equivalence relation.

Hence, the correct option is A.