In mathbb{R}, a relation p is defined as foll | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The relation is defined as $a^2-4ab+3b^2=0$. $1$. Reflexivity: For any $a \in \mathbb{R}$, we check if $a \ p \ a$. Substituting $b=a$, we get $a^

Vedclass · Accepted Answer

$p$ is only reflexive

Vedclass · Answer

(C) The relation is defined as $a^2-4ab+3b^2=0$. $1$. Reflexivity: For any $a \in \mathbb{R}$, we check if $a \ p \ a$. Substituting $b=a$, we get $a^2-4a(a)+3a^2 = a^2-4a^2+3a^2 = 0$. Since $0=0$, the relation is reflexive. $2$. Symmetry: Check if $a \ p \ b \implies b \ p \ a$. If $a^2-4ab+3b^2=0$, then $(a-b)(a-3b)=0$, so $a=b$ or $a=3b$. If $a=3b$, then $b=a/3$. For symmetry, we need $b^2-4ba+3a^2=0$. Substituting $a=3b$, we get $b^2-4b(3b)+3(3b)^2 = b^2-12b^2+27b^2 = 16b^2 
eq 0$ (unless $b=0$). Thus, it is not symmetric. $3$. Transitivity: Check if $a \ p \ b$ and $b \ p \ c \implies a \ p \ c$. If $a=3b$ and $b=3c$, then $a=9c$. For $a \ p \ c$, we need $a^2-4ac+3c^2=0$. Substituting $a=9c$, we get $(9c)^2-4(9c)c+3c^2 = 81c^2-36c^2+3c^2 = 48c^2 
eq 0$. Thus, it is not transitive. Therefore, the relation is only reflexive.

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

Similar Questions

The number of reflexive relations on a set $A$ of $n$ elements is equal to

Let $A = \{1, 2, 3, 4\}$ and let $R = \{(2, 2), (3, 3), (4, 4), (1, 2)\}$ be a relation on $A$. Then $R$ is

Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as $R_1 = \{(x,y) \in N \times N : 2x + y = 10\}$ and $R_2 = \{(x,y) \in N \times N : x + 2y = 10\}$. Then

Let $A = \{1, 2, 3\}$. Show that the number of relations on $A$ containing $(1, 2)$ and $(2, 3)$ which are reflexive and transitive but not symmetric is $3$. (Note: The original prompt claimed $4$,but the correct count for this specific set is $3$).

The relation "congruence modulo $m$" is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module