Consider the relation R on the set {-2, -1, 0 | vedclass

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

Question

(A) Let the set be $S = \{-2, -1, 0, 1, 2\}$. The total number of possible ordered pairs $(a, b)$ is $5 	imes 5 = 25$.We need to find pairs such that

Vedclass · Accepted Answer

Only $I$ is true

Vedclass · Answer

(A) Let the set be $S = \{-2, -1, 0, 1, 2\}$. The total number of possible ordered pairs $(a, b)$ is $5 	imes 5 = 25$.We need to find pairs such that $1 + ab > 0$, which is equivalent to $ab > -1$.It is easier to count the pairs where $ab \leq -1$ and subtract from $25$.The pairs $(a, b)$ where $ab \leq -1$ are:$(-2, 1) \Rightarrow ab = -2$$(1, -2) \Rightarrow ab = -2$$(-2, 2) \Rightarrow ab = -4$$(2, -2) \Rightarrow ab = -4$$(-1, 1) \Rightarrow ab = -1$$(1, -1) \Rightarrow ab = -1$$(-1, 2) \Rightarrow ab = -2$$(2, -1) \Rightarrow ab = -2$There are $8$ such pairs. Thus, the number of elements in $R$ is $25 - 8 = 17$. Statement $I$ is true.For Statement $II$, check if $R$ is an equivalence relation. $A$ relation is an equivalence relation if it is reflexive, symmetric, and transitive.Reflexivity: $1 + a^2 > 0$ is true for all $a \in S$. So, it is reflexive.Symmetry: $1 + ab > 0 \iff 1 + ba > 0$. So, it is symmetric.Transitivity: Consider $(1, 0) \in R$ because $1 + 0 = 1 > 0$, and $(0, -2) \in R$ because $1 + 0 = 1 > 0$. However, $(1, -2) 
otin R$ because $1 + (1)(-2) = -1 
gtr 0$. Thus, $R$ is not transitive. Statement $II$ is false.

Consider the relation $R$ on the set $\{-2, -1, 0, 1, 2\}$ defined by $(a, b) \in R$ if and only if $1 + ab > 0$. Then, among the statements:
$I$. The number of elements in $R$ is $17$
$II$. $R$ is an equivalence relation

Similar Questions

Let $P(S)$ denote the power set of $S = \{1, 2, 3, \ldots, 10\}$. Define the relations $R_1$ and $R_2$ on $P(S)$ as $A R_1 B$ if $(A \cap B^c) \cup (B \cap A^c) = \varnothing$ and $A R_2 B$ if $A \cup B^c = B \cup A^c, \forall A, B \in P(S)$. Then:

Show that the relation $R$ in the set $Z$ of integers given by $R = \{(a, b) : 2 \text{ divides } a - b\}$ is an equivalence relation.

On the set $N$ of natural numbers,the relation $R$ is defined by $nRm$ if $n$ is a factor of $m$ (i.e.,$n|m$). Then $R$ is:

Let $H$ be the set of all houses in a village where each house is faced in one of the directions,East,West,North,South. Let $R = \{ (x,y) | (x,y) \in H \times H \text{ and } x, y \text{ are faced in same direction} \}$. Then the relation $R$ is

Let $R$ be the relation in the set $\{1, 2, 3, 4\}$ given by $R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}$. Choose the correct answer.

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Consider the relation $R$ on the set $\{-2, -1, 0, 1, 2\}$ defined by $(a, b) \in R$ if and only if $1 + ab > 0$. Then, among the statements:$I$. The number of elements in $R$ is $17$$II$. $R$ is an equivalence relation