If `A={1,2,3,4},B={3,4,5},C={1,5,6}`, then findof the following :
(i) `A cup B`
(ii) `A cup (B cap C)`
(ii) `A - B`
(iv) `A - (B cup C)`.

Answer

Let's solve the problem step by step.

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

### (i) Find \( A \cup B \)

The union of two sets includes all the elements from both sets without repetition.

- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( B \): \( 3, 4, 5 \)

Combining these, we get:
\[ A \cup B = \{1, 2, 3, 4, 5\} \]

### (ii) Find \( A \cup (B \cap C) \)

First, we need to find \( B \cap C \) (the intersection of sets \( B \) and \( C \)), which includes only the elements that are present in both sets.

- Elements in \( B \): \( 3, 4, 5 \)
- Elements in \( C \): \( 1, 5, 6 \)

The common element is \( 5 \):
\[ B \cap C = \{5\} \]

Now, we find \( A \cup (B \cap C) \):
\[ A \cup (B \cap C) = A \cup \{5\} = \{1, 2, 3, 4, 5\} \]

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

The difference \( A - B \) includes elements that are in \( A \) but not in \( B \).

- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( B \): \( 3, 4, 5 \)

Removing \( 3 \) and \( 4 \) from \( A \):
\[ A - B = \{1, 2\} \]

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

First, we need to find \( B \cup C \) (the union of sets \( B \) and \( C \)).

- Elements in \( B \): \( 3, 4, 5 \)
- Elements in \( C \): \( 1, 5, 6 \)

Combining these, we get:
\[ B \cup C = \{1, 3, 4, 5, 6\} \]

Now, we find \( A - (B \cup C) \):
- Elements in \( A \): \( 1, 2, 3, 4 \)
- Elements in \( B \cup C \): \( 1, 3, 4, 5, 6 \)

Removing \( 1, 3, \) and \( 4 \) from \( A \):
\[ A - (B \cup C) = \{2\} \]

### Final Answers:
1. \( A \cup B = \{1, 2, 3, 4, 5\} \)
2. \( A \cup (B \cap C) = \{1, 2, 3, 4, 5\} \)
3. \( A - B = \{1, 2\} \)
4. \( A - (B \cup C) = \{2\} \)