Question

Let $X$ be the set of all persons living in a city. Persons $x,y$ in $X$ are said to be related as $x<y$ if $y$ is at least $5$ years older than $x$. Which one of the following is correct?

• A. The relation is an equivalence relation on $X$
• B. The relation is transitive but neither reflexive nor symmetric
• C. The relation is reflexive but neither nor symmetric
• D. The relation is symmetric but neither transitive nor reflexive

Answer 1

The correct option is B The relation is transitive but neither reflexive nor symmetric

Given, $X$ is the set of persons living ins a city.

Relation(R) = {(x,y): x,y $\in$ $R$ $\&$ x<y if y is atleast $5$ years older than x}

For Reflexive: $(x,x)$ should $\in R$ for all $x\in X$

now, $x$ cannot be be $5$ years older than himself. So the relation is not reflexive.

For Symmetric: If $(x,y)\in R\Rightarrow (y,x)\in R$

$(x,y)\in R\Rightarrow y$ is at least $5$ years older than $x$.

$(y,x)\in R\Rightarrow x$ is at least $5$ years older than $y$. This contradicts the above statement. Hence the relation is not symmetric

For Transitive: If $(x,y)\in R$ and $(y,z)\in R\Rightarrow (x,z)\in R$.

$(x,y)\in R\Rightarrow y$ is at least $5$ years older than $x$.

$(y,z)\in R\Rightarrow z$ is at least $5$ years older than $y$.

Then, $(x,z)\in R\Rightarrow z$ is at least $5$ years older than $x$.

Since, $z$ is at least $10$ years older than $x$. The relation is transitive.