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. Step 1: Understand the definition of the set. The set is defined as {x | x is a natural number greater than 11}. This means it includes all natural numbers starting from 12 and onwards.
2. Step 2: Determine if the set is finite or infinite. Since there is no upper limit to the natural numbers greater than 11, the set is infinite.
3. Step 3: Check if 10 is an element of the set. Since the set includes only natural numbers greater than 11, 10 is not included in the set.
4. Step 4: Conclude that the set is infinite and does not contain the number 10.
5. Step 5: Summarize the findings: The set is infinite, and 10 is not an element of the set.

Key Concepts

Finite vs. Infinite Sets

A finite set contains a limited number of elements, while an infinite set has no bounds and continues indefinitely. For example, the set of all natural numbers is infinite, as it includes 1, 2, 3, and so on without end. Understanding whether a set is finite or infinite is crucial for determining its properties and the nature of its elements.

Natural Numbers

Natural numbers are the set of positive integers starting from 1 and increasing indefinitely (1, 2, 3, ...). They are used for counting and ordering. In the context of the given set, recognizing that natural numbers do not include zero or negative numbers is essential for determining membership within the set.

Set Membership

Set membership refers to whether a specific element belongs to a particular set. In this case, to determine if 10 is an element of the set {x | x is a natural number greater than 11}, one must evaluate if 10 meets the criteria defined by the set. Since 10 is not greater than 11, it is not a member of this set.