Assuming one resident only in each unit, there are 58 residents in the survey. Represent this as a large circle R containing three smaller interlocking circles we'll call B (bird owners), F (fish owners) and other pets (P). Circle R contains 8 zones (Zn):
- Only birds
- Only fish
- Only other pets (excluding birds and fish)
- Birds, fish but no other pets
- Birds, other pets, but no fish
- Fish, other pets, but no birds
- Birds, fish and other pets (10)
- No pets
The Venn diagram can show these zones. Z1 is the part of the B circle that has no intersections with circles F and P; similarly Z2 for the F circle and Z3 for the P circle. Where circles B, P and F overlap showing elements common to between each set, there are 3 zones trapped within each circle. You should be able to see that Z7 appears in each circle because it represents the set of residents having birds, fish and other pets (frogs, lizards, snakes), while the other zones Z4, Z5 and Z6 have only elements common to two sets. Z8 is the zone outside of all the interlocking circles inside the big circle R. This picture will help you to see what's going on in the equations, which determine how many residents, if any, are in each zone.
B=30=Z1+Z4+Z5+Z7; F=38=Z2+Z4+Z6+Z7; P=30=Z3+Z5+Z6+Z7; R=58=Z1+...+Z8.
18=Z4+Z7; 15=Z5+Z7. But Z7=10, so Z4=8; Z5=5. From this we can find Z1=30-(8+5+10)=7.
Z2+Z6=38-18=20; Z3+Z6=30-15=15; 58=7+(20-Z6)+(15-Z6)+8+5+Z6+10, so 58=65-Z6 and Z6=65-58=7. Now we can find Z2=13; Z3=8; Z8=58-(7+13+8+8+5+7+10)=0. There are no reports of any resident having no pets, so all residents have at least one pet.