Use set notation, and list all the elements of each set. {x | x is a natural number not greater than 4}

Verified step by step guidance

1. Understand the problem: We need to describe the set of all natural numbers that are not greater than 4 using set notation and list all the elements explicitly.
2. Recall that natural numbers are typically the positive integers starting from 1, so the natural numbers not greater than 4 are those numbers \( x \) such that \( x \in \mathbb{N} \) and \( x \leq 4 \).
3. Write the set in set-builder notation: \( \{ x \mid x \in \mathbb{N}, x \leq 4 \} \). This means the set of all natural numbers \( x \) where \( x \) is less than or equal to 4.
4. List all the elements of the set explicitly by identifying all natural numbers from 1 up to 4: \( \{ 1, 2, 3, 4 \} \).
5. Combine both parts: the set in set-builder notation and the explicit list of elements, which fully describes the set.

Key Concepts

Set Notation

Set notation is a way to describe a collection of elements using a rule or property. The notation {x | condition} means 'the set of all x such that the condition holds.' It helps define sets precisely without listing all elements explicitly.

Natural Numbers

Natural numbers are the set of positive integers starting from 1 (sometimes including 0, depending on context). They are used for counting and ordering. Understanding which numbers qualify as natural is essential for correctly listing elements in the set.

Inequalities and Set Restrictions

Inequalities like 'not greater than 4' restrict the elements included in a set. This means all elements must satisfy the condition x ≤ 4. Recognizing and applying such restrictions helps in accurately listing all elements of the set.