Question

Let$A$ be the set of the children in a family. The relation $‘x’$ is a brother of $‘y’$on $A$ is

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of these

Answer 1

The correct option is C

Transitive

Explanation for correct option:

Finding the relation:

Given $A$ be the set of the children in a family.

$R$ is the relation where $X\to Y$ $‘x’$ is a brother of $‘y’$on $A$.

we may understand $X$ is set of boys and $Y$ is set of either boys or girls

Option (C): Transitive

If$R$ is a relation on a set $A$, then $R$ is said to be Transitive for all $x,y,z\in A$, $\left(x,y\right)\in R$and $\left(y,z\right)\in R$$\Rightarrow$$\left(x,z\right)\in R$.

Let $a,b,c\in A$,$\left(a,b\right)\in R$and $\left(b,c\right)\in R$,then $\left(a,c\right)\in R$

Since,$\left(a,b\right)\in R$means $a$ is the brother of $b$. $a$ is boy.

$\left(b,c\right)\in R$means $b$ is the brother of $c$. $b$ is boy.

$\Rightarrow$$\left(a,c\right)\in R$means $a$ is the brother of $c$.

$R$ is Transitive.

Therefore, Option (C) is correct.

Explanation for the wrong options:

Option (A): Reflexive

If$R$ is a relation on a set $A$, then $R$ is said to be Reflexive for all $x\in A$, $\left(x,x\right)\in R$.

Let $a\in A$,$\left(a,a\right)\notin R$.

Since,$a$ is not a brother of $a$ . A person is not a brother of himself.

$R$ is not Reflexive.

Therefore, Option (A) is not correct.

Option (B): Symmetric.

If$R$ is a relation on a set $A$, then $R$ is said to be Symmetric. for all $x,y\in A$, $\left(x,y\right)\in R$$\Rightarrow$$\left(y,x\right)\in R$.

Let $a,b\in A$,$\left(a,b\right)\in R$, then $\left(b,a\right)\notin R$

Since $\left(a,b\right)\in R$means $a$ is the brother of $b$. $b$ maybe a girl or boy.

$\Rightarrow \left(b,a\right)\notin R$because $b$ maybe sister of $a$.

$R$ is Symmetric.

Therefore, Option (B) is not correct.

Option (D): None of these.

$R$ is transitive.

Therefore, Option (D) is not correct.

Hence, option (C) is the correct answer.