Question

Find set A={1, 2, 6, 10} B={3, 6, 9, 10, 11} C = {1, 2, 4, 7, 11}

Answer 1

If U = {1, 2, 3, …, 12} is the universal set, and

A = {1, 2, 6, 10}

B = {3, 6, 9, 10, 11}

C = {1, 2, 4, 7, 11}

then

(1) A U B is the set containing all elements from A and B,

A U B = {1, 2, 3, 6, 9, 10, 11}

(2) A ∩ B is the set of elements that are contained in both A and B,

A ∩ B = {6, 10}

(3) Unfortunately, A ∩ B U C is somewhat ambiguous. It could mean (A ∩ B) U C or A ∩ (B U C ). Then either

(A ∩ B) U C = {6, 10} U {1, 2, 4, 7, 11} = {1, 2, 4, 6, 7, 10, 11}

or

A ∩ (B U C ) = {1, 2, 6, 10} ∩ {1, 2, 3, 4, 6, 7, 9, 10, 11} = {1, 2, 6, 10}

The first interpretation is probably the intended one, since that essentially reads the set operations from left to right.

(4) A' U B is the union of A' and B, where A' is the complement of A, or all elements in U that are not in A. We have

A' = U - A = {3, 4, 5, 7, 8, 9, 11, 12}

and so

A' U B = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

(5) We have

A U C = {1, 2, 4, 6, 7, 10, 11}

so that

(A U C )' = U - (A U C ) = {3, 5, 8, 9, 12}

(6) We have

B' = U - B = {1, 2, 4, 5, 7, 8, 12}

and so

A ∩ B' = {1, 2}

(7) Using the complements found in (4) and (6), we have

A' U B' = {1, 2, 3, 4, 5, 7, 8, 9, 11, 12}

Alternatively, we can use the fact that

A' U B' = (A ∩ B)'

and since we know from (2) that A ∩ B = {6, 10}, we end up with the same result,

(A ∩ B)' = U - (A ∩ B) = {1, 2, 3, 4, 5, 7, 8, 9, 11, 12}

(8) We have

A U B U C = {1, 2, 3, 4, 6, 7, 9, 10, 11}

so that

(A U B U C )' = {5, 8, 12}