The four sets A, B, C, and D each have 400 elements. The intersection of any two of the sets has 115 elements. The intersection of any three of the sets has 53 elements. The intersection of all four sets has 28 elements. How many elements are there in the union of the four sets

the union of the four sets have 1094 elements

Step-by-step explanation:

denoting as N as the number of elements, since

N (A U B) = N (A) + N (B) - N (A ∩ B)

then

N (A U B U C) = N (A U B) + N (C) - N (A U B ∩ C) = N (A) + N (B) - N (A ∩ B) + N (C) - [ N (A∩C) + N (B ∩ C) - N (A ∩ B ∩ C) ]

then for the union of 4 sets, we have

N (A U B U C U D) = N (A) + N (B) + N (C) + N (D) - N (A ∩ B) - N (A ∩ C) - N (A ∩ D) - N (B ∩ C) - N (B ∩ D) - N (C ∩ D) + N (A ∩ B ∩ C) + N (A ∩ B ∩ D) + N (A ∩ C ∩ D) + N (B ∩ C ∩ D) - N (A ∩ B ∩ C ∩ D)

thus replacing values for the sets, union of 2 sets, union of 3 sets and union of 4 sets

N (A U B U C U D) = (4*400) - (6*115) + (4*53) - 28 = 1094 elements

then the union of the four sets have 1094 elements