QuestionFill in the Venn diagram using: $n(A)=30$ , $n(B)=38$ , $n(C)=23$ , $n(A \cap B)=12$ , $n(B \cap C)=14$ , $n(A \cap C)=5$ , $n(A \cap B \cap C)=2$ , $n(U)=71$ .

Studdy Solution

STEP 1

Assumptions1. The number of elements in set A is30. The number of elements in set B is383. The number of elements in set C is234. The number of elements in the intersection of sets A and B is125. The number of elements in the intersection of sets B and C is146. The number of elements in the intersection of sets A and C is57. The number of elements in the intersection of sets A, B, and C is8. The total number of elements in the universal set U is71

STEP 2

First, we need to find the number of elements in each set that are not in the intersection with any other set. For set A, we subtract the number of elements in the intersections involving set A from the total number of elements in set A.
$n(A \setminus (B \cup C)) = n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)$

STEP 3

Now, plug in the given values to calculate the number of elements in set A that are not in the intersection with any other set.
$n(A \setminus (B \cup C)) =30 -12 -5 +2$

STEP 4

Calculate the number of elements in set A that are not in the intersection with any other set.
$n(A \setminus (B \cup C)) =30 -12 - +2 =15$

STEP 5

Repeat the process for sets B and C to find the number of elements in each set that are not in the intersection with any other set.
For set B $n(B \setminus (A \cup C)) = n(B) - n(A \cap B) - n(B \cap C) + n(A \cap B \cap C)$ For set C $n(C \setminus (A \cup B)) = n(C) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$

STEP 6

Plug in the given values to calculate the number of elements in set B that are not in the intersection with any other set.
$n(B \setminus (A \cup C)) =38 -12 -14 +2$

STEP 7

Calculate the number of elements in set B that are not in the intersection with any other set.
$n(B \setminus (A \cup C)) =38 -12 -14 +2 =14$

STEP 8

Plug in the given values to calculate the number of elements in set C that are not in the intersection with any other set.
$n(C \setminus (A \cup B)) =23 -5 -14 +2$

STEP 9

Calculate the number of elements in set C that are not in the intersection with any other set.
$n(C \setminus (A \cup B)) =23 -5 -14 +2 =6$

STEP 10

Finally, to find the number of elements that are not in any of the sets A, B, or C, we subtract the number of elements in each set and their intersections from the total number of elements in the universal set U.
$n(U \setminus (A \cup B \cup C)) = n(U) - n(A) - n(B) - n(C) + n(A \cap B) + n(A \cap C) + n(B \cap C) - n(A \cap B \cap C)$

STEP 11

Plug in the given values to calculate the number of elements that are not in any of the sets A, B, or C.
$n(U \setminus (A \cup B \cup C)) =71 -30 -38 -23 + +5 +14 -$

STEP 12

Calculate the number of elements that are not in any of the sets A, B, or C.
$n(U \setminus (A \cup B \cup C)) =71 -30 -38 -23 +12 +5 +14 -2 =9$ The numbers to fill in the Venn diagram are as follows- The number of elements only in set A15- The number of elements only in set B14- The number of elements only in set C6- The number of elements in the intersection of sets A and B but not C10- The number of elements in the intersection of sets B and C but not A12- The number of elements in the intersection of sets A and C but not B- The number of elements in the intersection of sets A, B, and C2- The number of elements not in any of the sets A, B, or C9

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