I have an exercise in the book (Probability and statistics for Engineering and the Sciences J.L. Devore) for which there is no answer in the key, so I would like to ask somebody to check it. And also I have no idea how do I create a Venn diagram to this one.
An engineering construction firm is currently working on power plants at three different sites. Let $A_i$ denote the event that the plant at site $i$ is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in the terms of $A_1$, $A_2$, $A_3$, draw a Venn diagram, and shade the region corresponding to each one.
a. At least one plant is completed by the contract date
b. All plants are completed by the contract date
c. Only the plant at site 1 or both of the other two plants are completed by the contract date
d. Exactly one plant is completed by the contract date
e. Either the plant at site $1$ or both of the other two plants are completed by the contract date
I have assumed that events are independent (changed from mutually exclusive). Then:
a. $\{A_1,A_2',A_3'\} \cup \{A_1',A_2,A_3'\} \cup \{A_1',A_2',A_3\}\cup \{A_1,A_2,A_3'\}\cup\{A_1,A_2',A_3\}\cup\{A_1',A_2,A_3\}\cup\{A_1,A_2,A_3\}$
b. $\{A_1, A_2, A_3\}$
c. $\{A_1, A_2′, A_3′\}$
d. $\{A_1, A_2′, A_3′\}\cup\{A_1′, A_2, A_3′\}\cup\{A_1′, A_2′, A_3\}$
e. $\{A_1, A_2′, A_3′\} \cap \left(\{A_1′, A_2, A_3′\} \cup \{A_1′, A_2′, A_3\}\right)$
EDIT: I have found a solution as amWhy has given. But still I think there's something wrong with the logic of this task. In my opinion the result of the "experiment" is 3-tuple and the space contains all 3-tuples contnaining all outcomes from $\{A_1,A_2,A_3\}$ to $\{A_1',A_2',A_3'\}$ then let's say $C=\{A_1,A_2',A_3'\}$ which means that the first plant is completed in time. And compound events as C can be used to draw Venn diagram.
Best Answer
The events are not mutually exclusive, but I'm assuming for this task, you are to take the events as independent.
For example: For the first, it suffices to write $A_1 \cup A_2 \cup A_3$. For the second, we have intersection $A_1 \cap A_2\cap A_3$. Do you understand why?
"Or" is usually taken to be "inclusive": denoted by union. $A$ or $B$ or both $A$ and $B$.
"And" is usually taken to be intersection.
If "exclusive or" is specified, so that we have $A$ or $B$ but not both $A$ and $B$, you need to use both union and intersection into account (for example, in $c$).
Then compare how $(c)$ and $(e)$ differ: $(c)$ is a bit more complicated: $(e)$ is simply asserting an inclusive or: $$(e)\quad A_1 \cup (A_2 \cap A_3)$$
whereas $(c)$ is exclusive, so we need to say $$(c)\quad \Big( A_1 \cap (A_2 \cap A_3)'\Big) \cup \Big(A_1' \cap (A_2 \cap A_3)\Big)$$
Can you go back and try to complete tasks? In particular, see if you can write $(d)$ using the given set operations.
With respect to Venn Diagrams, you only need paper and pencil to draw Venn Diagrams. Google "Venn Diagrams" and you'll find some "on-line" software where you can experiment with drawing the appropriate Venn Diagrams.