Let R be a relation on the set of all natural | vedclass

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

Question

(A) The relation is defined as $aRb \iff a \mid b^2$ for $a, b \in \mathbb{N}$.$I.$ Reflexivity: For any $a \in \mathbb{N}$,$a^2$ is always divisible

Vedclass · Accepted Answer

$I$ only

Vedclass · Answer

(A) The relation is defined as $aRb \iff a \mid b^2$ for $a, b \in \mathbb{N}$.$I.$ Reflexivity: For any $a \in \mathbb{N}$,$a^2$ is always divisible by $a$ (since $a^2/a = a \in \mathbb{N}$). Thus,$(a, a) \in R$ for all $a \in \mathbb{N}$. So,$R$ is reflexive.$II.$ Symmetry: For $R$ to be symmetric,$aRb \implies bRa$. Let $a=8$ and $b=4$. Here $8 \mid 4^2$ $(8 \mid 16)$ is true,so $(8, 4) \in R$. However,$4 \mid 8^2$ $(4 \mid 64)$ is true,but consider $a=2, b=4$. $2 \mid 4^2$ $(2 \mid 16)$ is true,but $4 \mid 2^2$ $(4 \mid 4)$ is true. Let us take $a=2, b=6$. $2 \mid 6^2$ $(2 \mid 36)$ is true,but $6 \mid 2^2$ $(6 \mid 4)$ is false. Thus,$R$ is not symmetric.$III.$ Transitivity: For $R$ to be transitive,$(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$. Let $a=8, b=4, c=2$. $(8, 4) \in R$ because $8 \mid 4^2$ $(8 \mid 16)$.$(4, 2) \in R$ because $4 \mid 2^2$ $(4 \mid 4)$.Check $(8, 2)$: $8 \mid 2^2$ $(8 \mid 4)$ is false. Thus,$R$ is not transitive.Therefore,only $I$ is satisfied.

Let $R$ be a relation on the set of all natural numbers $\mathbb{N}$ defined by $aRb \iff a \text{ divides } b^2$. Which of the following properties does $R$ satisfy?
$I.$ Reflexivity
$II.$ Symmetry
$III.$ Transitivity

Similar Questions

Let $A = \{1, 2, 3, 4\}$ and $R$ be a relation on the set $A \times A$ defined by $R = \{((a, b), (c, d)) : 2a + 3b = 4c + 5d\}$. Then the number of elements in $R$ is:

The number of reflexive relations on a set with $4$ elements is equal to

Check whether the relation $R$ defined in the set $\{1, 2, 3, 4, 5, 6\}$ as $R = \{(a, b) : b = a + 1\}$ is reflexive,symmetric,or transitive.

In the set of natural numbers,the relation "is less than" is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Let $R$ be a relation on the set of all natural numbers $\mathbb{N}$ defined by $aRb \iff a \text{ divides } b^2$. Which of the following properties does $R$ satisfy?$I.$ Reflexivity$II.$ Symmetry$III.$ Transitivity