If U = {3, 4, 5, 6, 7,8, 9} X = {3, 4}, Y = {5, 6} and Z = {7, 8, 9} then `Y' cap (X cap Z)'` is equal to

Options

A. `X cup Y`
B. `Y cup Z`
C. `(X cup Z)'`
D. `X' cap Y'`

Answer

To solve the problem, we need to find \( Y' \cap (X \cap Z)' \).

### Step 1: Identify the Universal Set and the Sets
Given:
- Universal Set \( U = \{3, 4, 5, 6, 7, 8, 9\} \)
- Set \( X = \{3, 4\} \)
- Set \( Y = \{5, 6\} \)
- Set \( Z = \{7, 8, 9\} \)

### Step 2: Find the Complement of Set Y
The complement of set \( Y \) (denoted as \( Y' \)) is found by subtracting the elements of \( Y \) from the universal set \( U \).

\[ Y' = U - Y = \{3, 4, 5, 6, 7, 8, 9\} - \{5, 6\} = \{3, 4, 7, 8, 9\} \]

### Step 3: Find the Intersection of Sets X and Z
Next, we need to find the intersection of sets \( X \) and \( Z \) (denoted as \( X \cap Z \)).

\[ X \cap Z = \{3, 4\} \cap \{7, 8, 9\} = \emptyset \]

### Step 4: Find the Complement of the Intersection
Now, we find the complement of the intersection \( (X \cap Z)' \).

Since \( X \cap Z = \emptyset \), its complement is:

\[ (X \cap Z)' = U - (X \cap Z) = U - \emptyset = U = \{3, 4, 5, 6, 7, 8, 9\} \]

### Step 5: Find the Intersection of Y' and (X ∩ Z)'
Now we need to find the intersection of \( Y' \) and \( (X \cap Z)' \).

\[ Y' \cap (X \cap Z)' = \{3, 4, 7, 8, 9\} \cap \{3, 4, 5, 6, 7, 8, 9\} \]

The common elements are:

\[ Y' \cap (X \cap Z)' = \{3, 4, 7, 8, 9\} \]

### Final Result
Thus, the final result is:

\[ Y' \cap (X \cap Z)' = \{3, 4, 7, 8, 9\} \]