Final answer:
Three students do not own either a cat or a dog after analyzing the distribution of pet ownership among 18 students using principles of set theory.
Step-by-step explanation:
The student's question pertains to the distribution of pet ownership among a sample of eighteen students, specifically regarding who owns cats, dogs, or neither. To solve this, we apply basic principles of set theory in mathematics to understand the overlap and distinct groups within the surveyed students. With 9 students having dogs, 6 having only cats, and 10 having cats (which includes those who have both cats and dogs), we can determine the number of students who have both cats and dogs by subtracting those who have only cats from the total number who have cats (10-6), giving us 4 students with both cats and dogs. Knowing this, we can find the total number of pet owners by adding those who have only dogs (9 students with dogs minus 4 who also have cats), which gives us 5 students with only dogs, adding to the 4 with both, and 6 with only cats, totalling 15 pet owners. Thus, 18 total surveyed students minus 15 pet owners gives us 3 students who do not have either a cat or a dog.