I need help with this Venn Diagram. I think I got the first (please check if they are right). The last one is the one that I'm missing. Would appreciate your help.
a. $A'∩B$
b. $(A∪B)'=A'∩B'$
c. $A∩B∩C$
d. $A∩B$
e. $(A∩C)-B$
f. ????
3 Answers
$\begingroup$Answers a) through d) are fine.
For e) I'm not sure an answer using the subtraction ("minus") symbol would be permitted, but it depends on your marking scheme. But you can always express that sort of answer in terms of a combination of union, intersection and complement.
So for e) I would write $A \cap C \cap B'$
For f) think of it like this:
You need the intersection of $A$ and $C$ for sure. But you also need the areas of $A$ that exclude $B$ and of $C$ that excludes $B$.
So you can construct it as $(A\cap C) \cup (A \cap B') \cup (C \cap B')$.
It's not a unique answer, expressions in set algebra often aren't.
$\endgroup$ 0 $\begingroup$Your answers for parts (a) to (e) are correct. For part (f), as mentioned in the comments, break the problem up into a few pieces and do it slowly. For example:
- Look at what the set $A\cup C$ represents in the Venn diagram.
- This is too much, and we need to remove two subsets of $B$, say $B_1,B_2$, to make it the desired set. You have already found the expression for subsets of this form in part (e).
Then the answer now is just $A\cup C-B_1-B_2$.
$\endgroup$ 3 $\begingroup$One of the things about these exercises is there is more than one way to describe a region in a Venn diagram.
For example, you know that (e) can be described as $(A\cap C) - B.$But it can also be written $A \cap C \cap B'.$
The simplest way to describe (d) is as you did: $A \cap B.$But you could also notice that the region is cut in two pieces by the boundary of $C$, and you could look at each of those pieces individually. One piece you have already worked out for (c): it is $A \cap B\cap C.$The other piece is like the one in (d) but it is shared by $A$ and $B$ rather than by $A$ and $C$, so you could write it as $(A \cap B) - C.$So another possible answer for (d) is $(A \cap B\cap C) \cup ((A \cap B) - C),$ although yours is arguably better since it does not require so much writing.
You can describe any region in a Venn diagram by breaking it into small enough pieces so that you know how to describe each piece separately. Then just take the union of the pieces you described, much as you could have taken the union$(A \cap B\cap C) \cup ((A \cap B) - C)$ for (d). This works even when there is not such a simple answer as there is for (d).
Another wrinkle is that when you break up your region into smaller regions in order to get regions you know how to describe, which you will then combine by taking a union, you don't really need the regions to be disjoint like the two separate regions in (d). It is OK if they overlap; since you are just taking a union of them in the end, all that matters is that each part of the shaded region is in at least one of the pieces you identified.
Yet another variation is that you can put together a region a little larger than the shaded region (perhaps using a union) and then take "bites" out of it by subtracting some sets.
You could use any of those techniques for a region like (f).
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