Question

Use Venn diagrams to determine whether the following statements are equal for all sets A, B, and C. 1. (AUB)’ n C 2. (A' UC') n (B' UC) Are the statements equal for all sets A, B, and C? a.No b. Yes

Answer 1

b. Yes

Step-by-step explanation:

The two statements (AUB)' n C and (A' UC') n (B' UC) are equal for all sets A, B, and C. This is because of De Morgan's Laws, which state that the complement of the union of two sets is equal to the intersection of their complements, and the complement of the intersection of two sets is equal to the union of their complements.

Let's break it down:

1. For the first statement, (AUB)' n C, applying De Morgan's Laws, the complement of the union of A and B is equal to the intersection of the complements of A and B. Then, the intersection of this result with set C is taken. This is exactly what the second statement (A' UC') n (B' UC) is expressing.

2. Therefore, both statements are essentially describing the same set operation, and they are equal for all sets A, B, and C.

In conclusion, the answer is "Yes" because the two statements are equivalent due to the consistent application of De Morgan's Laws, making them equal for any sets A, B, and C.

Venn diagram: