JAMIA 2024 — Mathematics PYQ

1. Reflexivity

A relation is reflexive if every element $a\in A$ is related to itself ($a,a)\in R$.

• For set $\{1,2,3\}$, $R$ must contain $(1,1),(2,2),$ and $(3,3)$.

• Since none of these are in $R=\{(1,2)\}$, the relation is not reflexive.

2. Symmetry

A relation is symmetric if whenever $(a,b)\in R$, then $(b,a)\in R$.

• In our case, $(1,2)\in R$.

• However, $(2,1)\notin R$.

• Therefore, the relation is not symmetric.

3. Transitivity

A relation is transitive if whenever $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$.

• To violate transitivity, we need a "chain" like $(a,b)$ and $(b,c)$ where the resulting $(a,c)$ is missing.

• In $R=\{(1,2)\}$, there is no second pair starting with $2$ to form a chain.

• According to logic, if the "if" part (the chain) doesn't exist, the statement is vacuously true.

• Therefore, the relation is transitive.

Final Answer

The relation $R=\{(1,2)\}$ is:

Transitive, but neither reflexive nor symmetric.