Show that each of relation R in set A={x ∈ Z: 0 ≤ x ≤ 12}, given by R ={( a , b ): a = b } is

English

Gujarati

Hindi

1.Relation and Function

medium

Show that each of the relation $R$ in the set $A=\{x \in Z: 0 \leq x \leq 12\},$ given by $R =\{( a , b ): a = b \}$ is an equivalence relation. Find the set of all elements related to $1$ in each case.

Option A

Option B

Option C

Option D

Solution

$R =\{( a , b ): a = b \}$

For any element a $\in A,$ we have $(a,\, a) \in R,$ since $a=a$

$\therefore R$ is reflexive.

Now, let $(a, b) \in R$

$\Rightarrow a=b$

$\Rightarrow b=a \Rightarrow(b, a) \in R$

$\therefore R$ is symmetric.

Now, let $(a, b) \in R$ and $(b, c) \in R$

$\Rightarrow a=b$ and $b=c$

$\Rightarrow a=c$

$\Rightarrow(a,\, c) \in R$

$\therefore R$ is transitive.

Hence, $R$ is an equivalence relation.

The elements in $R$ that are related to $1$ will be those elements from set $A$ which are equal to $1$

Hence, the set of elements related to $1$ is $\{1\}$.

Standard 12

Mathematics

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

medium

(JEE MAIN-2021)

Let $R\,= \{(x,y) : x,y \in N\, and\, x^2 -4xy +3y^2\, =0\}$, where $N$ is the set of all natural numbers. Then the relation $R$ is

hard

(JEE MAIN-2013)

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.

medium

Let $S=\{1,2,3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R=\{(A, B): A \cap B \neq \phi ; A, B \in M\}$ is :

hard

(JEE MAIN-2024)

Let $R$ and $S$ be two relations on a set $A$. Then

normal

Show that each of the relation $R$ in the set $A=\{x \in Z: 0 \leq x \leq 12\},$ given by $R =\{( a , b ): a = b \}$ is an equivalence relation. Find the set of all elements related to $1$ in each case.

Solution

Similar Questions

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

Let $R\,= \{(x,y) : x,y \in N\, and\, x^2 -4xy +3y^2\, =0\}$, where $N$ is the set of all natural numbers. 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.

Let $S=\{1,2,3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R=\{(A, B): A \cap B \neq \phi ; A, B \in M\}$ is :

Let $R$ and $S$ be two relations on a set $A$. Then

Start a Free Trial Now