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The relation R on the set Z of all integer numbers defined by

Question

The relation $R$ on the set $Z$ of all integer numbers defined by (x,y)\ \epsilon \ R\ \Leftrightarrow x-y is divisible by $n$ is

Equivalence

Symmetric only

Reflexive only

Transitive only

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Lucas Discussed

The relation

R

on the set

Z

of all integer numbers defined by (x,y)\ \epsilon \ R\ \Leftrightarrow x-y is divisible by

n

15 mins ago

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Text solutionVerified

R = {(x, y) : x, y \in z, x - y i s d i v i s ib l e b y n} F or R e f l e x i v e, x \in z S o \Rightarrow (x - x) i s d i v i s ib l e b y n \Rightarrow (x, x) \in z

So, Relation is Reflexive

F or S y mm e t r i c \Rightarrow L e t (x, y) \in R \Rightarrow (x - y) i s d i v i s ib l e b y n . \Rightarrow \frac{x - y}{n} = c, R e main d er i s 0. \Rightarrow \frac{y - x}{n} = - c, R e main d er i s a l so 0 . \Rightarrow (y - x) i s d i v i s ib l e b y n (y, x) \in R S o, R i s S y mm e t r i c . F or T r an s i t i v e \Rightarrow L e t (x, y) \in R & (y, z) \in R \Rightarrow (x - y) i s d i v i s ib l e b y n \Rightarrow (y - q) i s d i v i s ib l e b y n

add both these equation (i) & (ii)

(x - y) = x c, c \in z - v, (y - q) = na, (a \in z) \Rightarrow (x - y + y - q) = n c + na, c, a \in z \Rightarrow (x - q) = n (a + c), c, a \in z (x - q) i s d i v i s ib l e b y n \Rightarrow (x, q) \in R

So, R is Transitive

So, the R is Reflexive, Symmetric and Transitive
Then it is Equivalence Relation.

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Question Text	The relation $R$ on the set $Z$ of all integer numbers defined by (x,y)\ \epsilon \ R\ \Leftrightarrow x-y is divisible by $n$ is
Answer Type	Text solution:1
Upvotes	150