Is it possible to describe the empty set by using the venn diagram?
Also what is the venn diagram of a power set of a set with $n$ elements?
[Math] venn diagram of power set and empty set
elementary-set-theory
Related Solutions
You can express the various overlaps with $3$ circles, since there are three states for each of the companies (elements of $C$):
- Either it is in retail (an element of $R$) or it is not.
- Either it is quoted in the stock exchange (an element of $Q$) or it is not.
- Either it failed (is an element of $F$) or it is not.
Added: To start with, we have the following diagram:
We want to find that the maximum and minimum for that unknown value $x$. Since $5$ of the $90$ failed companies were not in retail, then $90-5=85$ were in retail. That is, of the studied companies, there were $85$ that failed and were retail companies. This tells us that $0\le x\le 85$ and that:
The two unfilled regions of $F$ must add up to $5$ (since $5$ of the failed companies were not in retail). Likewise, since $85$ of the $140$ retail companies failed, then $140-85=55$ of them did not, so the two unfilled regions of $R$ must add up to $55$. Hence:
where $0\le y\le 5$ and $0\le z\le 55.$ Now, all told, there are $120$ quoted companies, and we've accounted for $85-x+z+y$ of them so far. Thus:
Now, our total is $180+x-y-z,$ so:
Now, remember that $0\le x\le 85.$ We can refine this just a bit. Of the $200$ studied companies, $85$ of them were retail companies that failed and $120$ of them were companies quoted on the stock exchange. In order to make sure that there is "enough room" for these two sets of companies, we will need at least $120+85-200=5$ companies to be retail companies that failed and were quoted on the stock exchange, meaning $85-x\ge 5,$ meaning $80\ge x.$ Hence, our bounds on $x$ are $0\le x\le 80.$ At this point, we're all set to go. We know that $0\le x\le 80,$ so we have upper and lower bounds. To see whether they are the maximum and minimum, we must see if it is possible for those bounds to be attained (that is, see if it is possible for $x=80$ or $x=0$). The substitution $x=80$ gives us:
Recall that $0\le z\le 55.$ Of the $200$ companies, $140$ are in retail and $120$ were quoted, so to make sure there is "enough room" for those sets of companies, we need at least $140+120-200=60$ retail companies that were quoted. That is, $z+5\ge 60,$ so $z\ge 55.$ Hence, we need $z=55,$ and so:
Similarly, we need at least $90+120-200=10$ companies to be quoted companies that failed, meaning $5+y\ge10,$ so $y\ge 5,$ and since $0\le y\le 5,$ then $y=5,$ so:
Therefore, $80$ is the maximum number attainable for $x$.
Now, let's check and see if $x=0$ can be obtained. Substitution gives us:
Since $y\ge 0$ and $35-y-z\ge 0,$ then $35-z\ge 35-y-z\ge 0,$ so $z\le 35,$ so our new bounds of $z$ (having specified $x$) are $0\le z\le 35.$ Likewise, $y\le 35,$ but this doesn't refine the constraints $0\le y\le 5$ that we already had. Let's just pick a value of $z$ and see what happens. Say $z=10.$ Then:
That didn't introduce any issues, either, nor did it narrow our constraints for $y$ at all, so we can still pick any $0\le y\le 5$ and get an appropriate Venn diagram. Thus, $0$ is the minimum number attainable for $x$.
You can draw a dot (circle with area zero) anywhere in the Venn Diagram and that can sort of serve as an intuition for the empty set. It contains nothing and consequently, has no area.
In fact, you can draw many dots spread out all over the Venn Diagram like you spilled glitter on your diagram. It doesn't matter how many you draw or where you draw them, as long as they are area zero.
You'll see that the empty set is, indeed, a subset of every set $A$, because the statement: "For every element $e$ in the empty set, $e \in A$" is vacuously true.
Best Answer
The power set of a set with $n$ elements can be illustrated with a Venn diagram that has $n$ overlapping regions. The set with elments $1,2$ and $3$ but not $4$ or $5$ is represented by the region which is formed by the intersection of regions $1,2$, and $3$ with the intersections of the complements (exteriors) of $4$ and $5$, in a graph like this one that I found. The venn diagram of the empty set I'm not sure about. Maybe just a point.