Write the set using the roster method.

Y is the set of positive three-digit numbers where the first digit is smaller than the second by 2 and the third digit is smaller than the first by 4.

We are asked to write the set \( Y \) using the roster method. The elements of \( Y \) are positive three-digit numbers where:

- The first digit is smaller than the second digit by 2.
- The third digit is smaller than the first digit by 4.

Let the three-digit number be represented as \( \overline{abc} \), where:
- \( a \) is the first digit,
- \( b \) is the second digit, and
- \( c \) is the third digit.

From the problem's conditions, we have the following relationships:
1. \( b = a + 2 \) (the second digit is 2 more than the first),
2. \( c = a - 4 \) (the third digit is 4 less than the first).

The values of \( a \), \( b \), and \( c \) must satisfy the constraints for digits:
- \( a \) must be between 1 and 9 (since it's a three-digit number, the first digit cannot be 0),
- \( b \) and \( c \) must be valid digits (i.e., between 0 and 9).

Now, let's find the values of \( a \), \( b \), and \( c \) that satisfy these conditions:

### Step-by-step checking for valid \( a \), \( b \), and \( c \):
- If \( a = 6 \), then \( b = 8 \) and \( c = 2 \). This is valid.
- If \( a = 5 \), then \( b = 7 \) and \( c = 1 \). This is valid.
- If \( a = 4 \), then \( b = 6 \) and \( c = 0 \). This is valid.

For values of \( a \) less than 4, \( c \) becomes negative, which is not allowed.

Thus, the valid values for \( \overline{abc} \) are:
\[ Y = \{462, 571, 682\} \]

This is the set \( Y \) using the roster method.

Do you want more details or have any questions?

Here are 5 related questions:
1. What is the difference between the roster method and the set-builder notation?
2. How can we generalize the solution for different relationships between the digits?
3. What are other examples of sets that can be described using similar digit constraints?
4. Can the set \( Y \) be described using inequalities or other mathematical expressions?
5. How would the solution change if we imposed additional restrictions on the digits?

**Tip:** When solving digit-based problems, it's helpful to check the range of each digit separately to avoid invalid numbers.

Write the set using the roster method. Y is the set of positive three-digit numbers where the first digit is smaller than the second by 2 and the third digit is smaller than the first by 4.

Learn how to find the set of positive three-digit numbers where the first digit is smaller than the second by 2 and the third digit is smaller than the first by 4, using the roster method. Suitable for students in Grades 6-8, this problem explores set theory and number systems with step-by-step analysis.

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