If `A = {a, b, c} and B = {b, c, d, e}` be two subsets of the universal set `U = { a, b, c, d, e}`, then `( A cup B)`' is equal to

Options

A. `A ' cap B'`
B. `A' cup B'`
C. `A cup B'`
D. `A ' - B'`

Answer

To solve the problem, we need to find the complement of the union of sets A and B. Let's break down the steps:

### Step 1: Identify the sets
We have:
- Set A = {a, b, c}
- Set B = {b, c, d, e}
- Universal set U = {a, b, c, d, e}

### Step 2: Find the union of sets A and B
The union of two sets A and B, denoted as \( A \cup B \), includes all the elements that are in A, in B, or in both.

Calculating \( A \cup B \):
- Elements in A: {a, b, c}
- Elements in B: {b, c, d, e}

Combining these, we get:
\[ A \cup B = \{a, b, c, d, e\} \]

### Step 3: Find the complement of the union
The complement of a set \( X \), denoted as \( X' \), includes all the elements in the universal set U that are not in X.

Since \( A \cup B = \{a, b, c, d, e\} \), we now find the complement:
\[ (A \cup B)' = U - (A \cup B) \]

Calculating the complement:
- Universal set U = {a, b, c, d, e}
- Union \( A \cup B = \{a, b, c, d, e\} \)

Thus:
\[ (A \cup B)' = \{a, b, c, d, e\} - \{a, b, c, d, e\} = \emptyset \]

### Final Answer
The complement of \( A \cup B \) is:
\[ (A \cup B)' = \emptyset \]