Use set notation, and list all the elements of each set. {x | x is a natural number greater than 8 and less than 15}

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Identify the set notation given: \(\{ x \mid x \text{ is a natural number greater than 8 and less than 15} \}\).

Recall that natural numbers are positive integers starting from 1, so we are looking for integers \(x\) such that \(8 < x < 15\).

List all natural numbers that satisfy this inequality: these are the numbers 9, 10, 11, 12, 13, and 14.

Write the set in roster form by listing all these elements explicitly: \(\{ 9, 10, 11, 12, 13, 14 \}\).

Confirm that all elements are natural numbers and satisfy the original condition, completing the description of the set.

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주요 개념

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Set Notation

Set notation is a mathematical way to describe a collection of elements. It often uses curly braces {} and a rule or condition to specify which elements belong to the set, such as {x | condition on x}. This notation helps clearly define sets without listing all elements explicitly.

Natural Numbers

Natural numbers are the set of positive integers starting from 1, 2, 3, and so on. They are commonly used for counting and ordering. Understanding that natural numbers do not include zero or negative numbers is essential when identifying elements in a set.

Inequalities and Interval Constraints

Inequalities describe the range of values that elements in a set can take, such as 'greater than 8 and less than 15.' Interpreting these constraints correctly allows you to list all elements that satisfy the condition, which is crucial for writing the set explicitly.