Question

Finish labeling the number of elements in the regions in the Venn diagram shown to the right, where the subsets A, B, and C of the universe, U, satisfy the following conditions.

Answer 1

we have that

n(U)=100

so

(q+r+10+6)+(r+v)+w+x=100 ------> equation 1

n(A)=50

so

(q+r+10+6)=50 -------> equation 2

n(B)=40

so

(10+6+v+s)=40 -------> equation 3

n(C)=30

so

(r+6+v+w)=30 -------> equation 4

`n(A\cap B)=16`
`n(B\cap C)=14`

so

(6+v)=14 -------> v=14-6 --------> v=8

`n(A\cap C)=13`

so

r+6=13 --------> r=13-6 -------> r=7

step 1

substitute equation 2 in equation 1

50+(r+v)+w+x=100 ------> equation 5

we have

(10+6+v+s)=40 -------> equation 3

(r+6+v+w)=30 -------> equation 4

50+(r+v)+w+x=100 ------> equation 5

substitute the values of v and r in equation 4,5 and 6

(10+6+8+s)=40 --------> s=40-24 -------> s=16

equation 4

(7+6+8+w)=30 -------> W=30-21 -------> W=9

equation 5

50+(7+8)+9+x=100 -------> x=100-74 ------->x=26