Let ` U ` = the set of all triangles , P = the set of all isosceles triangles, Q - the set of all equilateral triangles , R = the set of all right angled triangles. What do the sets ` P uu Q ` represents ?

Explanation

To solve the question, we need to determine what the union of the sets \( P \) and \( Q \) represents. Let's break it down step by step.

### Step 1: Understand the Sets
- **Set \( U \)**: This is the set of all triangles.
- **Set \( P \)**: This is the set of all isosceles triangles, which are triangles with at least two equal sides.
- **Set \( Q \)**: This is the set of all equilateral triangles, which are triangles where all three sides are equal.
- **Set \( R \)**: This is the set of all right-angled triangles, which are triangles that have one angle equal to 90 degrees.

### Step 2: Define the Union of Sets
The union of two sets, denoted as \( P \cup Q \), includes all elements that are in either set \( P \) or set \( Q \) or in both.

### Step 3: Analyze the Relationship Between \( P \) and \( Q \)
- Every equilateral triangle (set \( Q \)) is also an isosceles triangle (set \( P \)) because it has at least two equal sides.
- However, not every isosceles triangle is equilateral, as an isosceles triangle can have two equal sides and a different third side.

### Step 4: Determine the Union \( P \cup Q \)
Since \( Q \) (equilateral triangles) is a subset of \( P \) (isosceles triangles), the union \( P \cup Q \) will include:
- All isosceles triangles (from set \( P \))
- All equilateral triangles (from set \( Q \))

Thus, the union \( P \cup Q \) represents the set of all isosceles triangles, which includes equilateral triangles as well.

### Final Answer
The set \( P \cup Q \) represents the set of all isosceles triangles.

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