n(T) = 150, n(C) = 225, n(T intersection C) = 100. Students drinking at least one = n(T union C) = n(T) + n(C) - n(T intersection C) = 150 + 225 - 100 = 275. Students drinking neither = total - n(T union C) = 600 - 275 = 325.

neither = n(U) - n(A union B)

A union B = {2,3,4,5,6,7,8}, so (A union B)' = U - (A union B) = {1,9,10}. Now A' = {1,6,7,8,9,10} and B' = {1,2,3,9,10}, so A' intersection B' = {1,9,10}. Since (A union B)' = {1,9,10} = A' intersection B', De Morgan's law is verified.

(A union B)' = A' intersection B'

For f(x) to be real, the expression under the square root must be non-negative: 9 - x^2 >= 0, i.e. x^2 <= 9, so -3 <= x <= 3. Hence the domain is [-3, 3]. As x varies over this interval, 9 - x^2 ranges from 0 (at x = +/-3) to 9 (at x = 0), so sqrt(9 - x^2) ranges from 0 to 3. Hence the range is [0, 3].

sqrt() defined only for non-negative values