Let S=Set of points inside the square, T=set of points inside the triangle and C=the set of points inside the circle. If the triangle and circle intersect each other and are contained in a square. Then (i) `SnnTnnC=phi` (ii) `SuuTuuC=C` (iii) `SuuTuuC=S` (iv) `SuuT=SnnC`

Options

A. `SnnTnnC=phi`
B. `SuuTuuC=C`
C. `SuuTuuC=S`
D. `SuuT=SnnC`

Answer

To solve the problem, we need to analyze the relationships between the sets \( S \), \( T \), and \( C \) based on the given conditions. Let's go through the options step by step.

### Step 1: Understand the Sets
- \( S \): Set of points inside the square.
- \( T \): Set of points inside the triangle.
- \( C \): Set of points inside the circle.

### Step 2: Analyze the Given Conditions
- The triangle \( T \) and the circle \( C \) intersect each other.
- Both \( T \) and \( C \) are contained within the square \( S \).

### Step 3: Evaluate Each Option

#### Option (i): \( S \cap T \cap C = \phi \)
- This option states that the intersection of all three sets is empty.
- Since the triangle and circle intersect, there are points that belong to both \( T \) and \( C \). Hence, \( S \cap T \cap C \) cannot be empty.
- **Conclusion**: This option is **false**.

#### Option (ii): \( S \cup T \cup C = C \)
- This option states that the union of all three sets is equal to the set of points inside the circle.
- Since both the triangle and the square contain points that are not in the circle, this option cannot be true.
- **Conclusion**: This option is **false**.

#### Option (iii): \( S \cup T \cup C = S \)
- This option states that the union of all three sets is equal to the set of points inside the square.
- Since both the triangle \( T \) and the circle \( C \) are contained within the square \( S \), this is indeed true.
- **Conclusion**: This option is **true**.

#### Option (iv): \( S \cup T = S \cap C \)
- This option states that the union of the square and triangle is equal to the intersection of the square and circle.
- Since the triangle is part of the square, \( S \cup T \) will include all points in \( S \) and some points in \( T \), while \( S \cap C \) will only include points that are in both \( S \) and \( C \). Hence, this statement is not true.
- **Conclusion**: This option is **false**.

### Final Conclusion
The only correct option is (iii): \( S \cup T \cup C = S \).