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Write Set \( B = \{ y \mid y \text{ is a multiple of 3 and } 12 \leq y \leq 32 \} \) Using Roster Method
Mathematics
Middle School (Grade 7-9)
Question Content
Write the set using the roster method: \( B = \left\{ y \mid y \text{ is a multiple of 3 and } 12 \leq y \leq 32 \right\} \)
Correct Answer
\( B = \{12, 15, 18, 21, 24, 27, 30\} \)
Detailed Solution Steps
1
1. Recall the definition of a multiple of 3: A number \( y \) is a multiple of 3 if \( y = 3k \) for some integer \( k \).
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2. Identify the range for \( y \): \( 12 \leq y \leq 32 \) (inclusive).
3
3. Find the smallest multiple of 3 in this range: \( 12 \) (since \( 12 = 3 \times 4 \) and \( 12 \geq 12 \)).
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4. Generate subsequent multiples of 3 by adding 3 each time, checking if they remain ≤ 32:
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- \( 12 + 3 = 15 \) (≤ 32)
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- \( 15 + 3 = 18 \) (≤ 32)
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- \( 18 + 3 = 21 \) (≤ 32)
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- \( 21 + 3 = 24 \) (≤ 32)
9
- \( 24 + 3 = 27 \) (≤ 32)
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- \( 27 + 3 = 30 \) (≤ 32)
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- \( 30 + 3 = 33 \) (> 32, so stop here).
12
5. List all valid multiples of 3 in the range: \( 12, 15, 18, 21, 24, 27, 30 \).
13
6. Express set \( B \) using the roster method: \( B = \{12, 15, 18, 21, 24, 27, 30\} \).
Knowledge Points Involved
1
Roster Method
A way to represent a set by explicitly listing all its elements within curly braces, e.g., \( \{a, b, c\} \).
2
Multiples of a Number
A multiple of a number \( n \) is an integer that can be written as \( n \times k \) (where \( k \) is an integer), e.g., multiples of 3 include \( 3, 6, 9, \dots \).
3
Set-Builder Notation
A notation to define a set by describing its elements (e.g., \( \{x \mid \text{condition on } x\} \)), where \( \mid \) means 'such that'.
4
Interval of Integers
A range of integers (e.g., \( 12 \leq y \leq 32 \)) specifies the lower and upper bounds for the elements of the set.
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