Let U= {1, 2, 3, 4, 5, 6, 7, 8,9}, A={1, 2, 3, 4}, B = {2, 4, 6, 8) and C={3, 4, 5, 6}. Find
`(B-C)'`

Answer

To solve the problem of finding `(B - C)'`, we will follow these steps:

### Step 1: Identify the Sets
We have the following sets:
- Universal set \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- Set \( B = \{2, 4, 6, 8\} \)
- Set \( C = \{3, 4, 5, 6\} \)

### Step 2: Find the Difference \( B - C \)
The difference \( B - C \) includes all elements that are in \( B \) but not in \( C \).

- Elements of \( B \): \( \{2, 4, 6, 8\} \)
- Elements of \( C \): \( \{3, 4, 5, 6\} \)

Now we will remove the elements of \( C \) from \( B \):
- \( 2 \) is in \( B \) and not in \( C \) (keep it)
- \( 4 \) is in both \( B \) and \( C \) (remove it)
- \( 6 \) is in both \( B \) and \( C \) (remove it)
- \( 8 \) is in \( B \) and not in \( C \) (keep it)

Thus, we have:
\[ B - C = \{2, 8\} \]

### Step 3: Find the Complement of \( B - C \)
Now we need to find the complement of \( B - C \) with respect to the universal set \( U \).

The complement \( (B - C)' \) includes all elements in \( U \) that are not in \( B - C \):
- \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
- \( B - C = \{2, 8\} \)

Now we will list the elements of \( U \) that are not in \( B - C \):
- \( 1 \) is not in \( \{2, 8\} \) (keep it)
- \( 2 \) is in \( \{2, 8\} \) (remove it)
- \( 3 \) is not in \( \{2, 8\} \) (keep it)
- \( 4 \) is not in \( \{2, 8\} \) (keep it)
- \( 5 \) is not in \( \{2, 8\} \) (keep it)
- \( 6 \) is not in \( \{2, 8\} \) (keep it)
- \( 7 \) is not in \( \{2, 8\} \) (keep it)
- \( 8 \) is in \( \{2, 8\} \) (remove it)
- \( 9 \) is not in \( \{2, 8\} \) (keep it)

Thus, the complement \( (B - C)' \) is:
\[ (B - C)' = \{1, 3, 4, 5, 6, 7, 9\} \]

### Final Answer
The final answer is:
\[ (B - C)' = \{1, 3, 4, 5, 6, 7, 9\} \]

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