If you want $x_1\neq x_2$, you can linearize $|x_1-x_2|\ge \varepsilon$, for example by introducing a boolean variable $y=1$ if and only if $x_1-x_2\ge \varepsilon$, and by imposing:
$$
x_1-x_2\le -\varepsilon +My\quad \mbox{and}\quad x_1-x_2\ge \varepsilon-(1-y)M
$$
Note: $\varepsilon$ is a "very small" constant close to zero and $M$ a very large integer.
This problem won't strictly be a linear program, but is better suited to Integer Programming.
First question: What sort of sets are working with?
We seem to have a set of jobs that must be completed. Additionally, we have a set of people who can do these jobs. Let's denote these sets $J$ and $P$, respectively. I'll assume those are the only sets you're dealing with, unless you'd like to clarify further.
Next question: What sort of decision variables will I need to formulate the problem?
Well, we probably need to assign people to jobs, in some fashion. Let $j\in J$ and $p\in P$ and let \begin{equation*}x_{ij}=\begin{cases}1& \text{job } j \text{ is completed by person }p\\
0& \text{otherwise}\end{cases}.\end{equation*}
Alternatively, you might want to assign a person to a group, the A and B you spoke of, or a group and a job, etc. You'll have to go back and make more sets, if that's the case.
What sort of parameters do you need? Maybe something that measures your utility of finishing one job before another? Then, you would need a parameter $v$ which encodes a value for each job.
What's your goal? Receive the most utility from completing these jobs? Complete as many as possible? etc. Encode whatever your objective is into a function using whatever sets, variables, and parameters you've defined above.
What constraints do you need? A common constraint in such a resource allocation problem might be that a person can only work on a finite number of jobs throughout the week, say $c$. We might write this constraint as
\begin{equation*}
\sum_{j\in J} x_{jp}\leq c \hspace{10pt}\forall p\in P.
\end{equation*}
The above info is a great general approach to formulating math programming problems.
As for your question regarding forbidding specific tasks from people who are in particular roles, let's suppose we have a set of roles $R$, jobs $J$, and people $P$. Consider $y_{rp}$, a decision variable equal to 1 if person $p$ is assigned to role $r$, and $x_{jp}$ from above. Further, let $J_r\subseteq J$ be a portion of the jobs only a member of role $r$ can perform. Then, we might write the constraint
\begin{equation*}
\sum_{j\in J'} x_{jp}\leq cy_{rp} \hspace{10pt}\forall p\in P.
\end{equation*}
Note that if person $p$ is not assigned to role $r$, he cannot perform any of these tasks, as
\begin{equation*}
\sum_{j\in J_r} x_{jp}\leq 0.
\end{equation*}
However, if he is assigned to role $r$, he may perform as many as $c$ tasks from subset $J_r$.
Does that help get you started? Feel free to ask other questions...
Best Answer
If the domain is not closed such as in the case of strict inequalities, the optimal value need not exists.
For example consider $\min x$ subject to $x>0$.
We can't find the smallest value as $x$ can get arbitrarily small and positive.
We can pick a small positive quantity and solve $\min x$ subject to $x \ge \epsilon$ instead if you desire a positive quantity but it is no longer the same problem.