Let R be a relation on set A of ordered pairs of positive integers defined by (x, y) R (u, v) if and

English

Gujarati

Hindi

1.Relation and Function

easy

Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $x v=y u .$ Show that $R$ is an equivalence relation.

Option A

Option B

Option C

Option D

Solution

Clearly, $(x, y)$ $R (x, y)$, $\forall \,\,(x, y) \in A$, since $x y=y x .$ This shows that $R$ is reflexive. Further, $(x, y) R (u, v)$ $ \Rightarrow x v=y u$ $ \Rightarrow u y=v x$ and hence $(u, v) \,R (x, y) .$ This shows that $R$ is symmetric. Similarly, $(x, y) R (u, v)$ and $(u, v)$ $R$ $(a, b) \Rightarrow x v=y u$ and $u b=v a \Rightarrow $ $x v \frac{a}{u}=y u \frac{a}{u} $ $\Rightarrow x v \frac{b}{v}=$ $y u \frac{a}{u} \Rightarrow $ $x b=y a$ and hence $(x, y) \,R (a, b) .$ Thus, $R$ is transitive. Thus, $R$ is an equivalence relation.

Standard 12

Mathematics

Let $R _{1}=\{( a , b ) \in N \times N 😐 a – b | \leq 13\}$ and $R _{2}=\{( a , b ) \in N \times N 😐 a – b | \neq 13\} .$ Thenon $N$

hard

(JEE MAIN-2022)

Show that the relation $R$ in the set $\{1,2,3\}$ given by $R =\{(1,1),\,(2,2),$ $(3,3)$, $(1,2)$, $(2,3)\}$ is reflexive but neither symmetric nor transitive.

easy

Let $A$ be a set consisting of $10$ elements. The number of non-empty relations from $A$ to $A$ that are reflexive but not symmetric is

normal

(KVPY-2020)

Let $A =\{1,2,3,4,5,6,7\}$. Then the relation $R =$ $\{( x , y ) \in A \times A : x + y =7\}$ is

medium

(JEE MAIN-2023)

Let $N$ denote the set of all natural numbers and $R$ be the relation on $N \times N$ defined by $(a, b)$ $R$ $(c, d)$ if $ad(b + c) = bc(a + d),$ then $R$ is