Determine the domain and range for the relation.

S={(x,y)||x∈ℤandy∈ℤ}

The relation \( S = \{(x, y) \mid x \in \mathbb{Z} \text{ and } y \in \mathbb{Z}\} \) represents the set of all ordered pairs \( (x, y) \) where both \( x \) and \( y \) are integers.

### Domain:
The domain of this relation is the set of all possible \( x \)-values. Since \( x \) can be any integer, the domain is:

\[
\text{Domain} = \mathbb{Z} = \{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \}
\]

### Range:
The range of this relation is the set of all possible \( y \)-values. Since \( y \) can also be any integer, the range is:

\[
\text{Range} = \mathbb{Z} = \{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \}
\]

### Conclusion:
- **Domain**: \( \mathbb{Z} \) (the set of all integers)
- **Range**: \( \mathbb{Z} \) (the set of all integers)

Would you like further details or have any questions?

#### Related Questions:
1. How do you find the domain and range of a general relation?
2. What is the difference between a relation and a function?
3. Can a relation have the same domain and range?
4. How do you represent relations graphically?
5. What happens if either \( x \) or \( y \) is restricted in a relation?

**Tip**: For a function to exist, every \( x \)-value must map to exactly one \( y \)-value. In a relation, this is not always required.

Determine the domain and range for the relation S = {(x,y) | x ∈ ℤ and y ∈ ℤ}

Understand how to determine the domain and range for a relation where both x and y are integers. This problem covers important concepts like relations, sets, and the domain and range of integers. Ideal for students in Grades 9-12.

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