Q.Statement (Q): $f : \mathbb{R} \to \mathbb{R}$ is a function defined as $f(x) = [x]$, greatest integer function, $f(x)$ is not onto function. Reason (R): A function $F : X \to Y$ is one-one if $F(a) = F(b) \Rightarrow a = b$. Alternatives:

(Q) is true (the floor function is not onto), and (R) is a true statement about one-one maps — but (R) explains injectivity, not the surjectivity claim in (Q), so it is not the correct explanation.

Onto (surjective) vs one-one (injective) functions is a CBSE/NCERT Class 12 relations and functions distinction.

(Q): $f(x)=[x]$ (greatest integer function) maps $\mathbb{R}\to\mathbb{R}$, but its range is only the set of integers $\mathbb{Z}$, which is a proper subset of $\mathbb{R}$. So $f$ is not onto — (Q) is true.

(R): The statement "$F$ is one-one if $F(a)=F(b)\Rightarrow a=b$" is the correct definition of a one-one (injective) function — true.

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