List the elements in each set. See Example 1. {z|z is a natural number greater than 4}
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Understand the problem: We need to list all elements of the set defined by the condition \(z\) is a natural number greater than 4.
Recall the definition of natural numbers: Natural numbers are the set of positive integers starting from 1, i.e., \(\{1, 2, 3, 4, 5, \ldots\}\).
Apply the condition 'greater than 4' to the natural numbers: This means we consider all natural numbers \(z\) such that \(z > 4\).
Write the elements of the set starting from the smallest natural number greater than 4, which is 5, and continue listing the numbers in increasing order: \(\{5, 6, 7, 8, \ldots\}\).
Since the set includes all natural numbers greater than 4 without an upper bound, the set is infinite and can be expressed as \(\{z \in \mathbb{N} \mid z > 4\} = \{5, 6, 7, 8, \ldots\}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Set Notation
Set notation is a way to describe a collection of elements that share a common property. In this question, the set is defined using a condition inside braces, indicating all elements z that satisfy the given property.
Natural numbers are the set of positive integers starting from 1 (1, 2, 3, ...). Understanding this helps identify which numbers qualify as elements of the set, especially when conditions like 'greater than 4' are applied.
Inequalities specify the range of values included in a set. Here, 'greater than 4' means all natural numbers strictly larger than 4 are included, so elements start from 5 onwards.