`{(a,x),(a,y),(b,x),(b,y)}` is a product set.

Answer

To determine whether the given set `{(a,x),(a,y),(b,x),(b,y)}` is a product set (Cartesian product), we will follow these steps:

### Step 1: Understand the Definition of Cartesian Product
The Cartesian product of two sets A and B, denoted as \( A \times B \), is defined as the set of all ordered pairs (a, b) where \( a \) is an element of set A and \( b \) is an element of set B. This means that each element in the Cartesian product is an ordered pair.

### Step 2: Identify the Elements in the Given Set
The given set is `{(a,x),(a,y),(b,x),(b,y)}`. We can see that it contains the following ordered pairs:
- (a, x)
- (a, y)
- (b, x)
- (b, y)

### Step 3: Determine the Sets from the Ordered Pairs
From the ordered pairs, we can identify two distinct sets:
- The first elements of the ordered pairs are {a, b}.
- The second elements of the ordered pairs are {x, y}.

Thus, we can define:
- Set A = {a, b}
- Set B = {x, y}

### Step 4: Calculate the Cartesian Product of Sets A and B
Now, we will compute the Cartesian product \( A \times B \):
\[ A \times B = \{(a, x), (a, y), (b, x), (b, y)\} \]

### Step 5: Compare the Given Set with the Cartesian Product
The calculated Cartesian product \( A \times B \) is exactly the same as the given set `{(a,x),(a,y),(b,x),(b,y)}`.

### Conclusion
Since the given set matches the Cartesian product of the two identified sets, we can conclude that the statement is true. Therefore, the given set is indeed a product set.