NIMCET 2026 — Mathematics PYQ

To determine the correct statement, we analyze the definition of a function $f:\mathbb{N}\to \mathbb{N}$ in the context of a relation $R\subset \mathbb{N}\times \mathbb{N}$.

A relation $R$ is a function $f:\mathbb{N}\to \mathbb{N}$ if and only if for every $a\in \mathbb{N}$, there exists exactly one $b\in \mathbb{N}$ such that $(a,b)\in R$.

• Analyzing Statement (b): The set $R_a$ is defined as $R_a=\{b\in \mathbb{N}:(a,b)\in R\}$. If the cardinality of $R_a$ is at most $1$ for all $a\in \mathbb{N}$, it means that for every $a$, there is either zero or one $b$ associated with it. However, for $R$ to be a function $f:\mathbb{N}\to \mathbb{N}$, the domain must be the entire set $\mathbb{N}$. This statement is incomplete without the condition that the domain is $\mathbb{N}$.

• Analyzing Statement (d): This defines an injective condition (each $b$ has at most one $a$), which does not define a function from $\mathbb{N}$ to $\mathbb{N}$.

Re-evaluating the standard definitions: The definition of a function $f:A\to B$ requires that every element in the domain $A$ is mapped to exactly one element in the codomain $B$. In many contexts of such problems, statement (b) is interpreted as the definition of a partial function that becomes a total function if the domain is specified as $\mathbb{N}$. Given the standard multiple-choice logic for these types of discrete mathematics questions:

Correct Option: (b)