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We are given two sets:

- \( S = \{1, 2, 3\} \)
- \( T = \{-2, -1, 0\} \)

The relation \( R \) is defined such that for each pair \( (x, y) \in S \times T \), \( (x - y) / 3 \) must be an integer. 

Let's find how many pairs satisfy this condition by going through each element of \( S \) and \( T \):

### Step 1: Pair all elements of \( S \times T \)
The possible pairs \( (x, y) \) are:
- \( (1, -2), (1, -1), (1, 0) \)
- \( (2, -2), (2, -1), (2, 0) \)
- \( (3, -2), (3, -1), (3, 0) \)

We will now check which pairs make \( \frac{x - y}{3} \) an integer.

### Step 2: Evaluate for each pair
1. \( (1, -2) \): \( \frac{1 - (-2)}{3} = \frac{1 + 2}{3} = \frac{3}{3} = 1 \), which is an integer.
2. \( (1, -1) \): \( \frac{1 - (-1)}{3} = \frac{1 + 1}{3} = \frac{2}{3} \), not an integer.
3. \( (1, 0) \): \( \frac{1 - 0}{3} = \frac{1}{3} \), not an integer.
4. \( (2, -2) \): \( \frac{2 - (-2)}{3} = \frac{2 + 2}{3} = \frac{4}{3} \), not an integer.
5. \( (2, -1) \): \( \frac{2 - (-1)}{3} = \frac{2 + 1}{3} = \frac{3}{3} = 1 \), which is an integer.
6. \( (2, 0) \): \( \frac{2 - 0}{3} = \frac{2}{3} \), not an integer.
7. \( (3, -2) \): \( \frac{3 - (-2)}{3} = \frac{3 + 2}{3} = \frac{5}{3} \), not an integer.
8. \( (3, -1) \): \( \frac{3 - (-1)}{3} = \frac{3 + 1}{3} = \frac{4}{3} \), not an integer.
9. \( (3, 0) \): \( \frac{3 - 0}{3} = \frac{3}{3} = 1 \), which is an integer.

### Step 3: Count valid pairs
The valid pairs where \( \frac{x - y}{3} \) is an integer are:
- \( (1, -2) \)
- \( (2, -1) \)
- \( (3, 0) \)

Thus, there are **3** such pairs.

### Conclusion:
The number of elements in \( R \) is \( \boxed{3} \).

Let me know if you need further clarifications or if you'd like more details. Here are 5 related questions for further practice:

1. What is the general formula for defining relations between two sets?
2. How would the answer change if \( S = \{1, 2, 3, 4\} \)?
3. How can you describe \( S \times T \) (Cartesian product) for two arbitrary sets?
4. What other integer conditions could we use to define relations between \( S \) and \( T \)?
5. How would the solution be different if the condition was \( (x + y) / 3 \)?

**Tip:** Always verify all possible pairs when analyzing relations based on a condition like divisibility to avoid missing any valid cases.

Let S = {1, 2, 3} and T = {-2, -1, 0}. Define a relation R from S to T as follows: For all (x, y) in S × T, (x, y) is in R means that (x - y) / 3 is an integer. Determine the number of elements of R.

This problem explores the relation R between two sets S and T, where the condition (x - y) / 3 must hold for the relation. A step-by-step solution is provided, checking each pair from the Cartesian product S × T to determine the valid elements in R. Suitable for students in Grades 8-10.

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