Venn Diagrams as Sets
*similar to problem 12 from section 1.2 of your text
Assume sets A and B are disjoint subsets of a universal set U, and assume that n(U) = 75, n(A) = 20, and n(B) = 35. Find n(A ∪ B)’.
I always recommend sketching these problems out as Venn diagrams, but in this case, what you'd end up with wouldn't actually be a Venn diagram since the two circles wouldn't overlap (if they're disjoint sets, then they have no intersection). And because they have no intersection, we know their union is simply the amount in each set added together: 20 + 35. That means A ∪ B = 55.
The complement of their union is the number of elements remaining in the universal set. In this case, there are 75 elements in U and we've accounted for 55 in A ∪ B, which must mean (A ∪ B)' = 20.
*similar to problem 29 from section 1.2 of your text
Six subsets X1, X2, X3, X4, X5, and X6 partition a set X such that n(X1) = n(X2) = n(X3) and n(X4) = n(X5) = n(X6). Assume that n(X1) = 4n(X4) and n(X) = 135. Find n(X1).
Problems like these can feel a bit abstract and difficult to conceptualize, so to make things easier, let's make a real world example out of it. Let's say I have $135 to give away (X) to the first six people who comment on my Instagram post (representing the six partitioned sets whose winnings will total exactly $135 -- no more, no less -- when added together). Except I'll give 4x as much money to the first three people as to the last three, so we might designate the winnings of the first person (X1) as 4x (or 4n) that of the fourth person (X4).
So how much money will the first person receive? Well, if the first three people each receive 4x the amount of the last three, then we can designate the winnings of the first three people as 4x + 4x + 4x and the last three as x + x + x (you can also use the variable n if you want to be consistent with the variable used in the question). So, together, that's 12x and 3x, and we know that when added, those winnings will total $135. Thus, we have 12x + 3x = 135, or 15x = 135, which means x = 9. Plug that into 4x, and the first person will receive $36. (You can check our math here by observing that 36 + 36 + 36 + 9 + 9 + 9 = 135.)
In mathematics terms, n(X1) = 36.
Let A and B be subsets of a universal set U, with n(U) = 100 and n(A ∪ B’) = 45. Determine the value of n(A’ ∩ B).
This problem is an excellent example of what I call the "complement rule." In this case, n(A ∪ B') and n(A' ∩ B) are complements because we took the complement of A and the complement of B' and flipped the union symbol to the intersection symbol, giving us (A' ∩ B). When you follow those steps, you always end up with the complement of the original subset. So the complement of (C' ∪ D') is (C ∩ D). The complement of (E ∩ F') is (E' ∪ F). Etc., etc.
That means if n(U) = 100 and (A ∪ B') and (A' ∩ B) are complements, then their values must total U when added together. Therefore, (A' ∩ B) = 55 because 45 + 55 = 100.
If you haven't quite mastered the complement rule, the long way for solving this problem would be to shade (A ∪ B') on your Venn diagram, which is everything in A, as well as everything not in B. In that case, you'll quickly see that the only thing you haven't shaded is that section of B that doesn't include its intersection with A -- in other words, everything in B that is also not in A, represented as (A' ∩ B). Now you know for sure that the two are complements and you can add/subtract as demonstrated above.
Given a universal set U with two subsets, A and B, which of the following statements is always true?
a. (A’ ∩ B) ⊂ (A ∪ B)’
b. (A ∪ B) ⊂ (A ∩ B’)
c. (A’ ∩ B) ⊂ (A ∩ B)’
d. (A ∩ B)’ ⊂ (A’ ∩ B’)
e. none of the above
With problems like these, there's no quick way to identify the correct answer. You just have to work through each answer one at a time until you find the right one.
So starting at the top, we must first understand what section of the Venn diagram is represented by (A' ∩ B). As discussed in Question 3, it's the section of B by itself (excluding its intersection with A). So is that a subset of the complement of A ∪ B? No, because (A ∪ B)' is everything not in A and/or B, and (A' ∩ B) is part of B.
By now, you probably know how to represent A ∪ B in your Venn diagram, which means we only need to determine if that's a subset of (A ∩ B'). In this case, one is definitely a part of the other, but it's flip-flopped. Because (A ∩ B') is the part of A by itself, that makes it a subset of A ∪ B, not the other way around. So this answer is also incorrect.
Referring back to answer a, we're already familiar with (A' ∩ B), the section of B by itself. Now we need to determine if that's a subset of (A ∩ B)'. To find that on the Venn diagram, shade everything that isn't the intersection of A and B. You'll see that what you've shaded also includes the section of B by itself represented by (A' ∩ B), which means our answer is c.
For the record, answer d works exactly like answer b. It's incorrect because it's flip-flopped. (A' ∩ B'), the section outside A ∪ B, is a subset of (A ∩ B)', not the other way around.