Let M={5,6,7,8} and N={3,4,9,10}. Which one of the following functions is neither one-one nor onto?

The correct option is:
$f=\left\{\right(5,3),(5,4),(6,4),(8,9\left)\right\}$

To understand why this relation/mapping is neither one-one nor onto, let us analyze its properties step-by-step:

1. Definition of a One-One (Injective) Mapping:
A mapping is one-one if distinct elements in the domain have distinct images in the codomain. That is, if
$a\ne b$
then their images must also be different.
Looking at the relation
$f=\left\{\right(5,3),(5,4),(6,4),(8,9\left)\right\}$,
we can see that the elements
$5$ and $6$
from the domain $M$ both map to the same element $4$ in the codomain $N$ (since we have the ordered pairs $(5,4)$ and $(6,4)$).
Since different inputs produce the same output, this mapping is not one-one.

2. Definition of an Onto (Surjective) Mapping:
A mapping is onto if every element in the codomain
$N=\{3,4,9,10\}$
has at least one pre-image in the domain $M$. In other words, the range of the mapping must be equal to the codomain.
Let us find the range (set of all image values) for
$f$:
$\text{Range}=\{3,4,9\}$
Comparing the range with the codomain $N$:
$\{3,4,9\}\ne \{3,4,9,10\}$
We observe that the element $10\in N$ has no pre-image in the domain $M$. Therefore, the mapping is not onto.

Furthermore, we can note that the element $7\in M$ has no assigned image, and the element $5\in M$ is mapped to multiple values ($3$ and $4$), meaning this relation does not qualify as a well-defined function. Among the given choices, it represents a mapping that is neither one-one nor onto.