Identify each set as finite or infinite. Then determine whether 10 is an element of the set. {x | x is a natural number greater than 11}

Verified step by step guidance

1. Understand the set notation: The set is defined as \( \{ x \mid x \text{ is a natural number greater than } 11 \} \). This means the set contains all natural numbers \( x \) such that \( x > 11 \).
2. Recall the definition of natural numbers: Natural numbers are the set \( \{1, 2, 3, 4, \ldots \} \), which continue infinitely.
3. Determine if the set is finite or infinite: Since the set includes all natural numbers greater than 11, it includes 12, 13, 14, and so on without end. Therefore, the set is infinite.
4. Check if 10 is an element of the set: Since 10 is a natural number but not greater than 11, it does not satisfy the condition \( x > 11 \). Hence, 10 is not an element of the set.
5. Summarize the findings: The set is infinite because it contains all natural numbers greater than 11, and 10 is not in the set because it does not meet the condition.

Key Concepts

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 natural number greater than 11), which means all natural numbers starting from 12 onward 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 no end. Since natural numbers greater than 11 continue indefinitely (12, 13, 14, ...), this set is infinite. Recognizing whether a set is finite or infinite helps in understanding its size and properties.

Element Membership in a Set

Determining if a specific number is an element of a set involves checking if it satisfies the set's defining condition. Here, 10 is not greater than 11, so it does not belong to the set. This concept is fundamental for verifying membership and solving related problems.