Q.Which of the following relations from $A=\{1,2,3\}$ to $B=\{p,q\}$ is a function?

A relation is a function only when every element of $A$ appears exactly once as a first component.

Options (a) and (d) repeat an element of $A$ with two images; option (b) leaves $3$ with no image; only (c) satisfies both existence and uniqueness.

For a relation from $A=\{1,2,3\}$ to $B=\{p,q\}$ to be a function, every element of $A$ must be the first component of exactly one ordered pair.

(a) $\{(1,p),(1,q),(2,p),(3,q)\}$: the element $1$ appears twice (with $p$ and with $q$) — two images, violating uniqueness. Not a function.

(b) $\{(1,p),(2,q)\}$: the element $3\in A$ has no image at all — violating existence. Not a function.

(c) $\{(1,q),(2,q),(3,q)\}$: each of $1,2,3$ appears exactly once, all mapped to $q$. Every element of $A$ has exactly one image, so this is a function (a constant function; it happens to be many-one and into, but it is still a function).

(d) $\{(1,p),(2,q),(3,p),(3,q)\}$: the element $3$ appears twice — two images. Not a function.

Only (c) satisfies the definition.

Students sometimes reject (c) because 'all inputs go to the same output', confusing this with 'one-one', or demand that both $p$ and $q$ be used, confusing it with 'onto'. Neither is required by the basic definition of a function.