Let R be the relation on N defined by x + 3y = 12, i.e. R = [(x, y) |x + 3y = 12).
(a) Write R as a set of ordered pairs. (c) Find R-*.
(b) Find the domain and range of R. (d) Find the composition relation RoR.

Let's break down the problem step by step for the given relation \( R \) defined on \( \mathbb{N} \) (the set of natural numbers) by the equation \( x + 3y = 12 \).

### (a) Write \( R \) as a set of ordered pairs.

The relation \( R \) consists of ordered pairs \( (x, y) \) such that \( x + 3y = 12 \) holds. Since \( x \) and \( y \) are natural numbers (\( \mathbb{N} \)), we need to find all such pairs that satisfy this equation.

Let's solve the equation \( x + 3y = 12 \) for natural number values of \( y \):

- For \( y = 0 \): \( x + 3(0) = 12 \Rightarrow x = 12 \).
- For \( y = 1 \): \( x + 3(1) = 12 \Rightarrow x = 9 \).
- For \( y = 2 \): \( x + 3(2) = 12 \Rightarrow x = 6 \).
- For \( y = 3 \): \( x + 3(3) = 12 \Rightarrow x = 3 \).
- For \( y = 4 \): \( x + 3(4) = 12 \Rightarrow x = 0 \).

Hence, the relation \( R \) as a set of ordered pairs is:

\[
R = \{ (12, 0), (9, 1), (6, 2), (3, 3), (0, 4) \}.
\]

### (b) Find the domain and range of \( R \).

- **Domain**: The domain of \( R \) is the set of all possible values of \( x \) that appear in the ordered pairs. From the set of ordered pairs \( R \), the domain is:
  \[
  \text{Domain of } R = \{ 12, 9, 6, 3, 0 \}.
  \]

- **Range**: The range of \( R \) is the set of all possible values of \( y \) that appear in the ordered pairs. From the set of ordered pairs \( R \), the range is:
  \[
  \text{Range of } R = \{ 0, 1, 2, 3, 4 \}.
  \]

### (c) Find \( R^{-1} \).

The inverse of \( R \), denoted \( R^{-1} \), is the set of ordered pairs obtained by swapping the elements of each pair in \( R \). So:

\[
R^{-1} = \{ (y, x) | (x, y) \in R \}.
\]

Thus, by swapping the elements of each pair in \( R \):

\[
R^{-1} = \{ (0, 12), (1, 9), (2, 6), (3, 3), (4, 0) \}.
\]

### (d) Find the composition relation \( R \circ R \).

The composition \( R \circ R \) means we are finding all pairs \( (x, z) \) such that there exists a \( y \) where \( (x, y) \in R \) and \( (y, z) \in R \).

We will go through the ordered pairs in \( R \):

1. \( (12, 0) \): There is no pair in \( R \) where the first element is 0, so no pair for \( x = 12 \).
2. \( (9, 1) \): There is no pair in \( R \) where the first element is 1, so no pair for \( x = 9 \).
3. \( (6, 2) \): There is no pair in \( R \) where the first element is 2, so no pair for \( x = 6 \).
4. \( (3, 3) \): The pair \( (3, 3) \) exists, so we have \( (3, 3) \in R \circ R \).
5. \( (0, 4) \): There is no pair in \( R \) where the first element is 4, so no pair for \( x = 0 \).

Thus, the composition relation \( R \circ R \) is:

\[
R \circ R = \{ (3, 3) \}.
\]

### Summary of Answers:

(a) \( R = \{ (12, 0), (9, 1), (6, 2), (3, 3), (0, 4) \} \).

(b) Domain of \( R = \{ 12, 9, 6, 3, 0 \} \), Range of \( R = \{ 0, 1, 2, 3, 4 \} \).

(c) \( R^{-1} = \{ (0, 12), (1, 9), (2, 6), (3, 3), (4, 0) \} \).

(d) \( R \circ R = \{ (3, 3) \} \).

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Do you have any questions or would you like more details?

Here are 5 related questions to consider:

1. How does the domain and range of a relation relate to the inverse of that relation?
2. What is the significance of the composition of two relations?
3. Can you provide an example where \( R \circ R \) includes more than one pair?
4. How would the relation change if the equation was \( x + 2y = 10 \)?
5. What would happen if the relation was defined over integers instead of natural numbers?

**Tip:** Inverse relations swap the roles of inputs and outputs. Always check how this affects both domain and range!

Let R be the relation on N defined by x + 3y = 12, i.e. R = [(x, y) |x + 3y = 12). (a) Write R as a set of ordered pairs. (b) Find the domain and range of R. (c) Find R-*. (d) Find the composition relation RoR.

This problem involves finding the set of ordered pairs that satisfy the equation x + 3y = 12, and exploring the domain, range, inverse relation, and composition. Perfect for students in Grades 10-12 studying algebraic relations and functions.

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