If `A ={6,7,8,9}, B={4,6,8,10}` and `C={x : x in N : 2 lt x le 7}`, find:
(i) A-B, (ii) B-C, (iii) B-(A-C), (iv) `A-(B cup C)`
(v) `B-(A cap C)`, (vi) B-B

Answer

To solve the question step by step, we will first define the sets given in the problem:

- Set A = {6, 7, 8, 9}
- Set B = {4, 6, 8, 10}
- Set C = {x : x in N, 2 < x ≤ 7} = {3, 4, 5, 6, 7} (since the natural numbers greater than 2 and less than or equal to 7 are 3, 4, 5, 6, and 7)

Now, we will solve each part of the question:

### (i) A - B
To find A - B, we remove the elements of B from A.

- A = {6, 7, 8, 9}
- B = {4, 6, 8, 10}

Removing 6 and 8 from A:
- A - B = {7, 9}

### (ii) B - C
To find B - C, we remove the elements of C from B.

- B = {4, 6, 8, 10}
- C = {3, 4, 5, 6, 7}

Removing 4 and 6 from B:
- B - C = {8, 10}

### (iii) B - (A - C)
First, we need to find A - C.

- A = {6, 7, 8, 9}
- C = {3, 4, 5, 6, 7}

Removing 6 and 7 from A:
- A - C = {8, 9}

Now, we find B - (A - C):
- B = {4, 6, 8, 10}
- A - C = {8, 9}

Removing 8 from B:
- B - (A - C) = {4, 6, 10}

### (iv) A - (B ∪ C)
First, we need to find B ∪ C.

- B = {4, 6, 8, 10}
- C = {3, 4, 5, 6, 7}

B ∪ C = {3, 4, 5, 6, 7, 8, 10}

Now, we find A - (B ∪ C):
- A = {6, 7, 8, 9}
- B ∪ C = {3, 4, 5, 6, 7, 8, 10}

Removing 6, 7, and 8 from A:
- A - (B ∪ C) = {9}

### (v) B - (A ∩ C)
First, we need to find A ∩ C.

- A = {6, 7, 8, 9}
- C = {3, 4, 5, 6, 7}

A ∩ C = {6, 7}

Now, we find B - (A ∩ C):
- B = {4, 6, 8, 10}
- A ∩ C = {6, 7}

Removing 6 from B:
- B - (A ∩ C) = {4, 8, 10}

### (vi) B - B
To find B - B, we remove all elements of B from B.

- B = {4, 6, 8, 10}

Removing all elements:
- B - B = ∅ (empty set)

### Summary of Results
1. A - B = {7, 9}
2. B - C = {8, 10}
3. B - (A - C) = {4, 6, 10}
4. A - (B ∪ C) = {9}
5. B - (A ∩ C) = {4, 8, 10}
6. B - B = ∅