TEZPUR 2025 — Mathematics PYQ

In set theory, a relation from set $A$ to set $B$ is defined as a subset of the Cartesian product $A\times B$.

First, we calculate the Cartesian product $A\times B$:

$$A\times B=\{(a,b):a\in A,b\in B\}$$

$$A\times B=\{(1,3),(1,4),(2,3),(2,4)\}$$

A relation $R$ must satisfy $R\subseteq A\times B$. Any ordered pair $(x,y)$ in the relation must have $x\in A$ and $y\in B$.

• (a) $\{(1,3),(1,4)\}\subseteq A\times B$ (Valid)

• (b) $\{(2,3),(2,4)\}\subseteq A\times B$ (Valid)

• (c) $\{(1,1),(1,2),(1,3),(1,4)\}$ contains elements $(1,1)$ and $(1,2)$. Since $1\notin B$ and $2\notin B$, these pairs are not in $A\times B$. (Invalid)

• (d) $\{(1,3),(1,4),(2,3),(2,4)\}=A\times B$ (Valid)

The correct answer is (c) because it contains ordered pairs that are not part of the Cartesian product of $A$ and $B$.