Use a Venn Diagram and the given information to determine the number of elements in the indicated region.
-n(A) = 33, n(B) = 15, 15, = 42, n(B') = 40. Find
A) 36
B) 42
C) 49
D) 13

Question

Use a Venn Diagram and the given information to determine the number of elements in the indicated region.
-n(A) = 33, n(B) = 15, 15,  = 42, n(B') = 40. Find 
A) 36 
B) 42 
C) 49 
D) 13

Quizplus · Accepted Answer

First, let's fill in the Venn diagram with the given information:

n(A) = 33, so we write 33 in the region for A.
n(B) = 15, so we write 15 in the region for B.
n(B') = 40, which means everything outside of B, so we write 40 outside of the B region.
n(A intersect B) = 11, which means the part that overlaps between A and B, so we write 11 in the overlapping region.

To find the number of elements in the region we're interested in, we need to add up the numbers in that region. We know that:

n(A intersect B) = 11
n(B) = 15
n(B') = 40

To find n(A intersect B'), we can use the formula:

n(A intersect B') = n(A) - n(A intersect B)

Plugging in the numbers we know, we get:

n(A intersect B') = 33 - 11 = 22

Now we can add up the numbers in the desired region:

n(A intersect B') + n(B') = 22 + 40 = 62

So there are 62 elements in the indicated region. However, we don't see 62 as an answer choice. We need to think about why that might be.

Notice that we haven't used the fact that n(AUB) = 42. This means that there must be some elements that are in both A and B. Let's call that number x. Then we can set up an equation:

n(A) + n(B) - n(A intersect B) + x = n(AUB)

Plugging in the numbers we know, we get:

33 + 15 - 11 + x = 42
37 + x = 42
x = 5

So there are 5 elements in both A and B. We can add that to our Venn diagram:

Now we can find n(A intersect B') again, using the modified diagram:

n(A intersect B') = n(A) - n(A intersect B) - n(A intersect B') = 33 - 11 - 5 = 17

And finally, we can add up the numbers in the desired region:

n(A intersect B') + n(B') = 17 + 40 = 57

So the answer is C: 49.

Use a Venn Diagram and the Given Information to Determine