Identify each set as finite or infinite. Then determine whether 10 is an element of the set. {x | x is a fraction between 8 and 9}

검증된 단계별 안내

Understand the set notation: The set is defined as \( \{ x \mid x \text{ is a fraction between 8 and 9} \} \), which means all fractions \( x \) such that \( 8 < x < 9 \).

Determine if the set is finite or infinite: Since there are infinitely many fractions between any two numbers, the set contains infinitely many elements, so it is an infinite set.

Check if 10 is an element of the set: Since 10 is greater than 9, it does not satisfy the condition \( 8 < x < 9 \), so 10 is not an element of the set.

Summarize the findings: The set is infinite, and 10 is not included in the set.

Note: Fractions include all rational numbers, so any rational number between 8 and 9 is part of the set.

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주요 개념

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Set Notation and Description

Set notation uses symbols to describe a collection of elements. In this question, the set is defined by a condition on x (x is a fraction between 8 and 9), meaning all fractions that satisfy this inequality are included. Understanding how to interpret such descriptions is essential for identifying the elements of the set.

Finite vs. Infinite Sets

A finite set has a limited number of elements, while an infinite set has unlimited elements. Since fractions between 8 and 9 can be infinitely many (there are infinitely many fractions between any two numbers), this set is infinite. Recognizing this helps classify the set correctly.

Element Membership in a Set

Determining if a number is an element of a set involves checking if it satisfies the set's defining condition. Here, to see if 10 is in the set, we check if 10 is a fraction between 8 and 9. Since 10 is greater than 9, it is not an element of the set.