Venn diagram consists of 4 circles: one large circle A enclosing three interlocking circles E, F, M.
A=all boys; E=English books; F=French books; M=mathematical books.
These circles divide space into 8 regions:
a. E only
b. F only
c. M only
d. E+F only
e. E+M only
f. F+M only
g. All
h. None
h=0 (no boy is without a book)
a+b+c+d+e+f+g=26.
a+d+e+g=19 (boys with English books, circle E regions)
b+d+f+g=23 (French, circle F regions); a+c+e=3
c+e+f+g=15 (circle M regions)
d+g=16; b+f=7
f+g=14; c+e=1, so c or e is zero and e or c equals 1; a=2; d+e+g=17; e=1 and c=0
e+g=13; c+f=2, so f=2 or 1; f=2, because c=0; g=12; b=5; d=4
Need to find:
- g=12 (boys with all 3 books)
- d+e+f=4+1+2=7 (two books only)
- d=4 (English and French, but not mathematics)
- b=23-(d+f+g)=23-18=5 (French books only)
- a+b+c=2+5+0=7 (only one book)
On the Venn diagram each circle consists of a region that doesn't overlap any other circle; two regions that each overlap one other circle; and a region that overlaps both of the other circles. Using the letters a to h the regions can be labelled and then filled with the values: (a,b,c,d,e,f,g,h)=(2,5,0,4,1,2,12,0). Examine each region and combination of regions and see how they apply to the question, particularly where figures have been supplied. Make sure the numbers in the regions correctly add up to these given numbers.