Answer to Question #270857 in Discrete Mathematics for Mary

Question

Answer what is asked.

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2, 4, 6, 8, 10}

B = {1, 3, 5, 7, 8, 9}

C = {2, 4, 6, 8, 10}

D = {4, 5, 6, 7, 8, 9, 10}

Find?

1.  A’ =

2.  B’ =

3.  C’ =

4.  D’ =

5.  A' ∪ B' =

6.  C' ∩ D' =

7. (A ∩ B) U C =

8.  A ∩ (B U D) =

9. (A ∩ D) - C =

10.(D' ∩  A') - B = 

Answer 1

1.

A' consist of elements in the universal set U but not in the set A,

Therefore, A'={3,5,7,9}

2.

B' consist of elements in the universal set U but not in the set B.

Therefore, B'={2,4,6,10}

3.

C' consist of elements in the universal set U but not in the set C.

Therefore, C'={1,3,5,7,9}

4.

D' consist of elements in the universal set U but not in the set D.

Therefore, D'={1,2,3}

5.

 A' ∪ B'  consist of all elements in A' and B'.

Therefore, A' ∪ B'={2,3,4,5,6,7,9,10}

6.

 C' ∩ D' consist of all elements common to C' and D'.

Therefore,  C' ∩ D' ={1,3}

7.

(A ∩ B) U C

For this part, we first determine A ∩ B.

A ∩ B consist of elements common to both set A and B.

Therefore, A ∩ B={1,8}

Given, A ∩ B={1,8} and C = {2, 4, 6, 8, 10} then, (A ∩ B) U C={1,2,4,6,8,10}

8.

To determine the elements of  A ∩ (B U D), we need to determine  B U D. B U D is the set of all elements in both sets B and D. Thus B U D={1,3,4,5,6,7,8,9,10}. Since A = {1, 2, 4, 6, 8, 10} then,  A ∩ (B U D)={1,4,6,8,10}.

Therefore, A ∩ (B U D)={1,4,6,8,10}.

9.

 (A ∩ D) - C.

We first need to determine the elements of A ∩ D. A ∩ D consist of elements common to both set A and D. Hence, A ∩ D={4,6,8,10} and C = {2, 4, 6, 8, 10}. (A ∩ D) - C consist of the elements in A ∩ D but are not in set C. Clearly, all elements in A ∩ D are in the set C. Therefore, (A ∩ D) - C is an empty set and is written as  (A ∩ D) - C=$\{\emptyset\}$.

10.

To find the elements in (D' ∩ A') - B, we first determine the elements in D' ∩ A' which are elements common to D' and A'. Thus, D' ∩ A'={3}. Now, (D' ∩ A') - B is defined as the elements in (D' ∩ A') that are not in B. Given that B = {1, 3, 5, 7, 8, 9} then it is clear that all elements in (D' ∩ A') are in set B. Therefore, (D' ∩ A') - B is empty and is written as,

(D' ∩ A') - B=$\{\emptyset\}$.