Question

A universal set U consists of 19 elements. If sets A, B, and C are proper subsets of U and n(U) = 19, n(An B) = n(An K C) = n(B n C)= 9, n(An B n C) =6, and n(A U B UC) = 15, determine each of the following. a) n(A U B) b ) n ( A' UC c) n(An B)'

Answer 1

Using Venn sets, the cardinalities are given as follows:

a) n(A U B) = 15.

b) n(A' U C) = 16.

c) n(A ∩ B)' = 10.

What are Venn probability?

Venn amounts relates the cardinality of sets that intersect with each other.

For this problem, the sets are the ones given in this problem, A, B and C, while U is the universal set.

For this problem, the cardinalities are given as follows:

• n(U) = 19.

• n(A ∩ B) = n(A ∩ C) = n(B ∩ C) = 9.

• n(A ∩ B ∩ C) = 6.

• n(A U B UC) = 15

Hence:

• 6 elements belong to all the sets.

• 9 - 6 = 3 belong to these intersections but not the remaining set: A and B, A and C, B and C.

• 15 belong to the union of all of them, hence 4 belong to none.

• 15 - (6 + 3 x 3) = 0 belong to only one set.

Hence:

• n(A U B) = 15, as from the final bullet point, there are no elements that belong to only set C.

• For item b, 6(all) + 3(only A and C) + 3 (only B and C) = 12 elements belong to C, and 4 do not belong to A(the 3 to only B and C is already counted), hence: n(A' U C) = 16, as 12 + 4 = 16.

• For item c, n(A ∩ B) = 9, hence n(A ∩ B)' = n(U) - n(A ∩ B) = 19 - 9 = 10.

More can be learned about Venn sets at https://brainly.com/question/28318748

#SPJ1