UPSC NDA 2 2021 Mathematics Solved Paper

Concept:A function is a special type of relation where each input has exactly one output.A relation between two sets is always a subset of their Cartesian product.Explanation:Statement 1: "All relations are functions but all functions are not relations."This is incorrect because every function is a relation, but not every relation is a function.A relation can have one input mapping to multiple outputs, which violates the function definition.Hence, statement 1 is false.Statement 2: "A relation from $A$ to $B$ is a subset of Cartesian product $A \times B$."This is correct.By definition, a relation from set $A$ to set $B$ is a subset of $A \times B$.So, statement 2 is true.Statement 3: "A relation in $A$ is a subset of Cartesian product $A \times A$."This is also correct.A relation on a set $A$ (i.e., from $A$ to $A$) is a subset of $A \times A$.The first element is the preimage and the second is the image.Thus, statement 3 is true.Answer:Only statements 2 and 3 are correct.So, the correct option is B.