UPSC NDA I 2015 Mathematics Paper

Concept:A relation $R$ on a set $X$ is:• Reflexive if every element is related to itself, i.e., $xRx$ for all $x \in X$.• Symmetric if $xRy$ implies $yRx$ for all $x,y \in X$.• Transitive if $xRy$ and $yRz$ imply $xRz$ for all $x,y,z \in X$.Explanation:The given relation: $x < y$ means $y$ is at least 5 years older than $x$.That is, $(x,y) \in R$ if and only if $y \ge x + 5$.Check reflexivity: For any person $x$, we need $x \ge x + 5$, which is impossible. So $R$ is not reflexive.Check symmetry: If $(x,y) \in R$, then $y \ge x+5$. For symmetry, we would need $(y,x) \in R$, i.e., $x \ge y+5$. This contradicts $y \ge x+5$ (unless both hold, which is impossible). Hence $R$ is not symmetric.Check transitivity: Suppose $(x,y) \in R$ and $(y,z) \in R$. Then $y \ge x+5$ and $z \ge y+5$. Combining gives $z \ge (x+5) + 5 = x+10$, so certainly $z \ge x+5$. Thus $(x,z) \in R$. Therefore $R$ is transitive.The relation is transitive but neither reflexive nor symmetric.Answer:Option B. The relation is transitive but neither reflexive nor symmetric.