QuestionFill in the Venn diagram with: $n(A)=27$ , $n(B)=34$ , $n(C)=27$ , $n(A \cap B)=10$ , $n(B \cap C)=10$ , $n(A \cap C)=10$ , $n(A \cap B \cap C)=2$ , $n(U)=71$ .

Studdy Solution

STEP 1

Assumptions1. The number of elements in set A, B, and C are given as27,34, and27 respectively. . The number of elements in the intersections of A and B, B and C, and A and C are given as10,10, and10 respectively.
3. The number of elements in the intersection of A, B, and C is given as.
4. The total number of elements in the universal set U is given as71.
5. The Venn diagram is divided into8 regions A only, B only, C only, A and B only, B and C only, A and C only, A, B and C, and outside of A, B, and C.

STEP 2

First, we need to find the number of elements that are only in A, B, and C. We can do this by subtracting the number of elements in the intersections from the total number of elements in each set.
For A only $n(A\, only) = n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)$

STEP 3

Now, plug in the given values to calculate the number of elements that are only in A.
$n(A\, only) =27 -10 -10 +2$

STEP 4

Calculate the number of elements that are only in A.
$n(A\, only) =27 -10 -10 +2 =9$

STEP 5

Repeat the process for B only and C only.
For B only $n(B\, only) = n(B) - n(A \cap B) - n(B \cap C) + n(A \cap B \cap C)$ For C only $n(C\, only) = n(C) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$

STEP 6

Plug in the given values to calculate the number of elements that are only in B and C.
$n(B\, only) =34 -10 -10 +2$ $n(C\, only) =27 -10 -10 +2$

STEP 7

Calculate the number of elements that are only in B and C.
$n(B\, only) =34 -10 -10 +2 =16$ $n(C\, only) =27 -10 -10 +2 =9$

STEP 8

Now, we need to find the number of elements that are in the intersections of A and B, B and C, and A and C but not in the intersection of A, B, and C.
For A and B only $n(A \cap B\, only) = n(A \cap B) - n(A \cap B \cap C)$ For B and C only $n(B \cap C\, only) = n(B \cap C) - n(A \cap B \cap C)$ For A and C only $n(A \cap C\, only) = n(A \cap C) - n(A \cap B \cap C)$

STEP 9

Plug in the given values to calculate the number of elements that are in the intersections of A and B, B and C, and A and C but not in the intersection of A, B, and C.
$n(A \cap B\, only) = -2$ $n(B \cap C\, only) = -2$ $n(A \cap C\, only) = -2$

STEP 10

Calculate the number of elements that are in the intersections of A and B, B and C, and A and C but not in the intersection of A, B, and C.
$n(A \cap B\, only) =10 -2 =8$ $n(B \cap C\, only) =10 -2 =8$ $n(A \cap C\, only) =10 -2 =8$

STEP 11

Finally, we need to find the number of elements that are outside of A, B, and C. We can do this by subtracting the number of elements in A, B, and C from the total number of elements in the universal set U.
$n(Outside\, A, B, C) = n(U) - n(A) - n(B) - n(C) + n(A \cap B) + n(B \cap C) + n(A \cap C) -n(A \cap B \cap C)$

STEP 12

Plug in the given values to calculate the number of elements that are outside of A, B, and C.
$n(Outside\, A, B, C) =71 -27 -34 -27 +10 +10 +10 -2*2$

STEP 13

Calculate the number of elements that are outside of A, B, and C.
$n(Outside\, A, B, C) =71 -27 -34 -27 +10 +10 +10 -2*2 =$ The numbers in the Venn diagram are as follows- A only9- B only16- C only9- A and B only8- B and C only8- A and C only8- A, B, and C2- Outside of A, B, and C

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