If `A = {1,2,3,4},B={2,4,6,8} and C={3,4,5,6}`, then find each of the following ,
(i) `(A - B) cap C`
(ii) `(A cap B) -C`
(iii) ` A- (B-C)`
(iv) `A cup (B - C)`.

Answer

To solve the given problem step by step, we will find the required operations on the sets \( A \), \( B \), and \( C \).

Given:
- \( A = \{1, 2, 3, 4\} \)
- \( B = \{2, 4, 6, 8\} \)
- \( C = \{3, 4, 5, 6\} \)

### (i) \( (A - B) \cap C \)

**Step 1:** Find \( A - B \)
To find \( A - B \), we remove the elements of \( B \) from \( A \).
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( B \): \( 2, 4, 6, 8 \)
- Removing \( 2 \) and \( 4 \) from \( A \):
\( A - B = \{1, 3\} \)

**Step 2:** Find \( (A - B) \cap C \)
Now we find the intersection of \( A - B \) with \( C \).
- Elements in \( A - B \): \( 1, 3 \)
- Elements in \( C \): \( 3, 4, 5, 6 \)
- The common element is \( 3 \):
\( (A - B) \cap C = \{3\} \)

### (ii) \( (A \cap B) - C \)

**Step 1:** Find \( A \cap B \)
To find \( A \cap B \), we look for common elements in \( A \) and \( B \).
- Common elements: \( 2, 4 \)
\( A \cap B = \{2, 4\} \)

**Step 2:** Find \( (A \cap B) - C \)
Now we subtract the elements of \( C \) from \( A \cap B \).
- Elements in \( A \cap B \): \( 2, 4 \)
- Elements in \( C \): \( 3, 4, 5, 6 \)
- Removing \( 4 \):
\( (A \cap B) - C = \{2\} \)

### (iii) \( A - (B - C) \)

**Step 1:** Find \( B - C \)
To find \( B - C \), we remove the elements of \( C \) from \( B \).
- Elements in \( B \): \( 2, 4, 6, 8 \)
- Elements in \( C \): \( 3, 4, 5, 6 \)
- Removing \( 4 \) and \( 6 \):
\( B - C = \{2, 8\} \)

**Step 2:** Find \( A - (B - C) \)
Now we subtract \( B - C \) from \( A \).
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Removing \( 2 \) and \( 8 \) (only \( 2 \) is in \( A \)):
\( A - (B - C) = \{1, 3, 4\} \)

### (iv) \( A \cup (B - C) \)

**Step 1:** Use the previously found \( B - C \)
From the previous step, we have:
\( B - C = \{2, 8\} \)

**Step 2:** Find \( A \cup (B - C) \)
Now we find the union of \( A \) and \( B - C \).
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( B - C \): \( 2, 8 \)
- The union will include all unique elements:
\( A \cup (B - C) = \{1, 2, 3, 4, 8\} \)

### Final Answers:
(i) \( (A - B) \cap C = \{3\} \)
(ii) \( (A \cap B) - C = \{2\} \)
(iii) \( A - (B - C) = \{1, 3, 4\} \)
(iv) \( A \cup (B - C) = \{1, 2, 3, 4, 8\} \)