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

A relation from $A$ to $B$ is a function only if every element of $A$ appears exactly once as a first component.

Options (a) and (d) repeat an element of $A$ with two different images, and option (b) leaves the element $3$ of $A$ with no image at all — only (c) avoids both problems.

For a relation from $A=\{1,2,3\}$ to $B=\{a,b\}$ to be a function, every element of $A$ must appear as the first component of exactly one ordered pair — not zero times, and not more than once.

(a) $\{(1,a),(1,b),(2,a),(3,b)\}$: the element $1$ appears twice, paired with both $a$ and $b$. Since $1$ has two different images, this is not a function.

(b) $\{(1,a),(2,b)\}$: the element $3 \in A$ does not appear as a first component at all — it has no image assigned. Since not every element of $A$ has an image, this is not a function (it is only a relation on the subset $\{1,2\}$ of $A$).

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

(d) $\{(1,a),(2,b),(3,a),(3,b)\}$: the element $3$ appears twice, paired with both $a$ and $b$. This is not a function, for the same reason as (a).

Only option (c) satisfies the defining condition — every element of $A$ related to exactly one element of $B$.

Students sometimes think a function must use every element of $B$ (confusing it with 'onto') or that repeating the same image for different inputs disqualifies it (confusing it with 'one-one') — neither of these is part of the basic definition of a function.