On the set of integers Z,a relation S is defi | vedclass

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

Question

(A) The condition $|x - y| < 1$ for integers $x, y \in Z$ implies that $|x - y| = 0$,because the absolute difference between two integers must be a no

Vedclass · Accepted Answer

It is an equivalence relation.

Vedclass · Answer

(A) The condition $|x - y| Thus,$|x - y| = 0 \implies x = y$. Therefore,$S = \{(x, x) : x \in Z\}$,which is the identity relation on $Z$. $1$. Reflexive: For any $x \in Z$,$|x - x| = 0 $2$. Symmetric: If $(x, y) \in S$,then $x = y$,which implies $y = x$,so $(y, x) \in S$. Thus,$S$ is symmetric. $3$. Transitive: If $(x, y) \in S$ and $(y, z) \in S$,then $x = y$ and $y = z$,which implies $x = z$,so $(x, z) \in S$. Thus,$S$ is transitive. Since $S$ is reflexive,symmetric,and transitive,it is an equivalence relation. Note: The provided option $B$ is incorrect based on the definition of the identity relation. The correct classification is an equivalence relation.

On the set of integers $Z$,a relation $S$ is defined as: $S = \{(x, y) \in Z \times Z : |x - y| < 1\}$. Which of the following is true about $S$?

Similar Questions

Let $R$ be a relation from $N$ to $N$ defined by $R = \{(a, b) : a, b \in N \text{ and } a = b^2\}$. Is the following statement true?
$(a, a) \in R$,for all $a \in N$

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?

Let $R = \{( P , Q ) \mid P \text{ and } Q \text{ are at the same distance from the origin} \}$ be a relation. Then the equivalence class of $(1, -1)$ is the set:

For a set $A = \{1, 2, 3\}$,a relation $R = \{(1, 2), (2, 3)\}$ is defined. What is the minimum number of ordered pairs that must be added to $R$ to make it an equivalence relation?

Vedclass Test Series

Exam Paper Generator

Online Exam Module