Use set-builder notation to write the following set.
{1, 2, 3, 4, 5, …}
The Correct Answer and Explanation is :
To express the set ({1, 2, 3, 4, 5, \ldots}) in set-builder notation, we can write it as:
[
S = {x \in \mathbb{Z}^+ \mid x \geq 1}
]
Explanation
Set-Builder Notation: Set-builder notation is a concise way to describe a set by stating the properties that its members must satisfy. It generally follows the form ({x \mid \text{property of } x}), where (x) is a variable that represents elements in the set, and the “property” is a condition that all elements (x) must meet to belong to the set.
Breaking Down the Notation:
- (S): This is the name of our set.
- (x): This variable represents the elements of the set. We can think of (x) as a placeholder for any member of the set we want to define.
- (\in \mathbb{Z}^+): This part specifies that (x) must be an element of the set of positive integers. (\mathbb{Z}^+) denotes the set of all positive integers, which includes 1, 2, 3, and so on. This ensures that we are only considering whole numbers greater than zero.
- (\mid): This symbol means “such that.” It separates the variable (x) from the property that defines the set.
- (x \geq 1): This condition states that (x) must be greater than or equal to 1. Thus, it includes 1 and all subsequent positive integers.
Infinite Set: The notation captures the infinite nature of the set. The ellipsis ((\ldots)) in the original set indicates that the integers continue indefinitely, which is preserved by stating the property (x \geq 1).
In conclusion, the set ({1, 2, 3, 4, 5, \ldots}) can be accurately described in set-builder notation as (S = {x \in \mathbb{Z}^+ \mid x \geq 1}), reflecting both its elements and their defining characteristics.