Let n be a fixed positive integer. The relati | vedclass

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

Question

(D) $1$. Reflexivity: For every $a \in Z$,$a - a = 0$. Since $n | 0$ is true for any positive integer $n$,$aRa$ holds. Thus,$R$ is reflexive.$2$. Symm

Vedclass · Accepted Answer

All of the above $(a), (b),$ and $(c)$

Vedclass · Answer

(D) $1$. Reflexivity: For every $a \in Z$,$a - a = 0$. Since $n | 0$ is true for any positive integer $n$,$aRa$ holds. Thus,$R$ is reflexive.$2$. Symmetry: If $aRb$,then $n | (a - b)$,which means $a - b = nk$ for some integer $k$. Then $b - a = -(a - b) = n(-k)$,which implies $n | (b - a)$. Thus,$bRa$ holds,and $R$ is symmetric.$3$. Transitivity: If $aRb$ and $bRc$,then $n | (a - b)$ and $n | (b - c)$. This implies $a - b = nk_1$ and $b - c = nk_2$ for some integers $k_1, k_2$. Adding these equations,$(a - b) + (b - c) = a - c = n(k_1 + k_2)$. Since $n | (a - c)$,$aRc$ holds,and $R$ is transitive.Conclusion: Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation. Therefore,the correct option is $(D)$.

Let $n$ be a fixed positive integer. The relation $R$ on the set of integers $Z$ is defined by $aRb \Leftrightarrow n | (a - b)$. Then $R$ is:

Similar Questions

If $A = \{x \in Z^+ : x < 10\}$ and $x$ is a multiple of $3$ or $4$,where $Z^+$ is the set of positive integers,then the total number of symmetric relations on $A$ is

Let $R$ be the real line. Let the relations $S$ and $T$ on $R$ be defined by $S = \{(x, y) : y = x + 1, 0 < x < 2\}$ and $T = \{(x, y) : (x - y) \text{ is an integer}\}$. Then:

Let $R$ be a transitive relation on a set $A$ and $I$ be the identity relation on $A$. Then:

Let $T$ be the set of all triangles in a plane with $R$ a relation in $T$ given by $R = \{(T_1, T_2) : T_1 \text{ is congruent to } T_2\}$. Show that $R$ is an equivalence relation.

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