Let n be a fixed positive integer. Define a r | vedclass

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

Question

(D) relation $R$ is defined on the set of integers $Z$ as $aRb \Leftrightarrow n | (a - b)$.$1$. Reflexive: For any $a \in Z$,$(a - a) = 0$. Since $n

Vedclass · Accepted Answer

All of the above

Vedclass · Answer

(D) relation $R$ is defined on the set of integers $Z$ as $aRb \Leftrightarrow n | (a - b)$.$1$. Reflexive: For any $a \in Z$,$(a - a) = 0$. Since $n | 0$ for any positive integer $n$,$aRa$ holds. Thus,$R$ is reflexive.$2$. Symmetric: If $aRb$,then $n | (a - b)$,which means $(a - b) = nk$ for some integer $k$. Then $(b - a) = -(a - b) = -nk = n(-k)$. Since $-k$ is an integer,$n | (b - a)$,so $bRa$. Thus,$R$ is symmetric.$3$. Transitive: 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,$(a - c) = (a - b) + (b - c) = n(k_1 + k_2)$. Since $(k_1 + k_2)$ is an integer,$n | (a - c)$,so $aRc$. Thus,$R$ is transitive.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. Define a relation $R$ on the set $Z$ of integers by $aRb \Leftrightarrow n | (a - b)$. Then $R$ is

Similar Questions

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 is given by:
$R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}$

Let $R = \{(a, a)\}$ be a relation on a set $A$. Then $R$ is:

Let a relation $R$ on $N \times N$ be defined as: $(x_1, y_1) R (x_2, y_2)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements:
$(I)$ $R$ is reflexive but not symmetric.
$(II)$ $R$ is transitive.
Then which one of the following is true?

If $R$ and $S$ are two non-empty relations on a set $A$,then which of the following statements is false?

$R = \{(\pi, \pi), (\pi^2, \pi^2), (\pi^3, \pi^3), (\pi, \pi^2), (\pi^2, \pi^3)\}$ is defined on the set $A = \{\pi, \pi^2, \pi^3\}$. Then $R$ is . . . . . . .

Vedclass Test Series

Exam Paper Generator

Online Exam Module