Given, A = {Triangles}, B = {Isosceles triangles}, C = {Equilateral triangles}. State whether the following are true or false. Give reasons.
(i) `A sub B`, (ii) `B sube A`, (iii) `C sube B`, (iv) `B sub A`, (v) `C sub A`, (vi) `C sube B sube A`

Answer

To determine whether the statements about the sets A, B, and C are true or false, we first need to understand the definitions of these sets:

- A = {Triangles}
- B = {Isosceles triangles}
- C = {Equilateral triangles}

Now, let's evaluate each statement step by step.

### Step 1: Evaluate `A ⊆ B`
**Statement:** A is a subset of B (A ⊆ B)
**Reasoning:** This statement is **False**. Not all triangles are isosceles triangles. For example, right-angled triangles and scalene triangles are also types of triangles that do not fall under the category of isosceles triangles.

### Step 2: Evaluate `B ⊆ A`
**Statement:** B is a subset of A (B ⊆ A)
**Reasoning:** This statement is **True**. Every isosceles triangle is indeed a triangle, so the set of isosceles triangles is contained within the set of all triangles.

### Step 3: Evaluate `C ⊆ B`
**Statement:** C is a subset of B (C ⊆ B)
**Reasoning:** This statement is **True**. Every equilateral triangle is also an isosceles triangle (since it has at least two sides that are equal), so the set of equilateral triangles is contained within the set of isosceles triangles.

### Step 4: Evaluate `B ⊆ A` (again)
**Statement:** B is a subset of A (B ⊆ A)
**Reasoning:** This statement is **True**. As previously mentioned, all isosceles triangles are triangles, confirming that B is indeed a subset of A.

### Step 5: Evaluate `C ⊆ A`
**Statement:** C is a subset of A (C ⊆ A)
**Reasoning:** This statement is **True**. All equilateral triangles are triangles, so the set of equilateral triangles is contained within the set of all triangles.

### Step 6: Evaluate `C ⊆ B ⊆ A`
**Statement:** C is a subset of B, and B is a subset of A (C ⊆ B ⊆ A)
**Reasoning:** This statement is **True**. Since we have established that C is a subset of B and B is a subset of A, it follows that C is also a subset of A.

### Summary of Results:
1. A ⊆ B: **False**
2. B ⊆ A: **True**
3. C ⊆ B: **True**
4. B ⊆ A: **True**
5. C ⊆ A: **True**
6. C ⊆ B ⊆ A: **True**