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Solved on Sep 17, 2023

Given a Venn diagram with 4 regions, find the number of elements in regions I, III, and IV given n(A)=16n(A)=16, n(B)=33n(B)=33, n(U)=54n(U)=54, and region II has 8 elements.
Region I has n(A)−n(A∩B)n(A) - n(A \cap B) element(s). Region III has n(B)−n(A∩B)n(B) - n(A \cap B) element(s). Region IV has n(U)−n(A)−n(B)+n(A∩B)n(U) - n(A) - n(B) + n(A \cap B) element(s).

STEP 1

Assumptions1. The Venn diagram has regions I, II, III, and IV. . The number of elements in region II is8.
3. The number of elements in set A, denoted by n(A)n(A), is16.
4. The number of elements in set B, denoted by n(B)n(B), is33.
5. The number of elements in the universal set U, denoted by n(U)n(U), is54.

STEP 2

First, we need to understand that the number of elements in set A and set B includes the elements in region II. Therefore, to find the number of elements in regions I and III, we need to subtract the number of elements in region II from the number of elements in sets A and B respectively.
For region I, we haven(I)=n(A)−n(II)n(I) = n(A) - n(II)For region III, we haven(III)=n(B)−n(II)n(III) = n(B) - n(II)

STEP 3

Now, plug in the given values for n(A)n(A), n(B)n(B), and n(II)n(II) to calculate the number of elements in regions I and III.
For region I, we haven(I)=16−8n(I) =16 -8For region III, we haven(III)=33−8n(III) =33 -8

STEP 4

Calculate the number of elements in regions I and III.
For region I, we haven(I)=16−8=8n(I) =16 -8 =8For region III, we haven(III)=33−8=25n(III) =33 -8 =25

STEP 5

Now, to find the number of elements in region IV, we need to understand that the number of elements in the universal set U includes the elements in all regions (, II, III, and IV). Therefore, to find the number of elements in region IV, we need to subtract the number of elements in regions I, II, and III from the number of elements in the universal set U.
n(IV)=n(U)−[n(I)+n(II)+n(III)]n(IV) = n(U) - [n(I) + n(II) + n(III)]

STEP 6

Now, plug in the given values for n(U)n(U), n(I)n(I), n(II)n(II), and n(III)n(III) to calculate the number of elements in region IV.
n(IV)=54−[8+8+25]n(IV) =54 - [8 +8 +25]

STEP 7

Calculate the number of elements in region IV.
n(IV)=54−[++25]=54−41=13n(IV) =54 - [ + +25] =54 -41 =13Region I has elements. Region III has25 elements. Region IV has13 elements.

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