Let R be a relation defined on N such that a | vedclass

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

Question

(D) $1$. Reflexivity: For any $a \in N$,$2a + 3a = 5a$,which is a multiple of $5$. Thus,$a R a$ holds for all $a \in N$. So,$R$ is reflexive.$2$. Symm

Vedclass · Accepted Answer

an equivalence relation

Vedclass · Answer

(D) $1$. Reflexivity: For any $a \in N$,$2a + 3a = 5a$,which is a multiple of $5$. Thus,$a R a$ holds for all $a \in N$. So,$R$ is reflexive.$2$. Symmetry: Let $a R b$,then $2a + 3b = 5k$ for some integer $k$. We want to check if $b R a$,i.e.,if $2b + 3a$ is a multiple of $5$.Note that $(2a + 3b) + (2b + 3a) = 5a + 5b = 5(a + b)$.Since $2a + 3b = 5k$,we have $2b + 3a = 5(a + b) - 5k = 5(a + b - k)$.Since $a, b, k$ are integers,$5(a + b - k)$ is a multiple of $5$. Thus,$b R a$ holds. So,$R$ is symmetric.$3$. Transitivity: Let $a R b$ and $b R c$. Then $2a + 3b = 5k_1$ and $2b + 3c = 5k_2$ for some integers $k_1, k_2$.We want to check if $a R c$,i.e.,if $2a + 3c$ is a multiple of $5$.From $2a + 3b = 5k_1$,we have $2a = 5k_1 - 3b$.From $2b + 3c = 5k_2$,we have $3c = 5k_2 - 2b$.Adding these,$2a + 3c = 5k_1 + 5k_2 - 5b = 5(k_1 + k_2 - b)$.Since $k_1, k_2, b$ are integers,$2a + 3c$ is a multiple of $5$. Thus,$a R c$ holds. So,$R$ is transitive.Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.

Let $R$ be a relation defined on $N$ such that $a R b$ if $2a + 3b$ is a multiple of $5$,where $a, b \in N$. Then $R$ is

Similar Questions

Let $A = \{2, 3, 6, 8, 9, 11\}$ and $B = \{1, 4, 5, 10, 15\}$. Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3ad - 7bc$ is an even integer. Then the relation $R$ is

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time given by $R = \{(x, y): x \text{ and } y \text{ work at the same place}\}$.

If $A = \{1, 2, 3, \dots, m\}$,then the total number of reflexive relations that can be defined from $A \to A$ is:

Let $R_{1}$ and $R_{2}$ be two relations defined on the set of real numbers $\mathbb{R}$ by $a R_{1} b \iff ab \geq 0$ and $a R_{2} b \iff a \geq b$. Then:

Let the relation $\rho$ be defined on $\mathbb{R}$ by $a \rho b$ if and only if $a-b$ is zero or irrational. Then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module