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Natural numbers are the simplest form of numbers we use every day. They are the numbers we start counting with, starting from 1 and increasing without any upper limit. Natural numbers are represented by the symbol \(\mathbf{N}\). Since these numbers include every number starting from 1 and increasing by 1 each time, they do not include zero, negative numbers, or any fractions or decimals.

In math, natural numbers are fundamental because they are the building blocks for other sets of numbers. They are essential for basic arithmetic operations like addition, subtraction, multiplication, and division, as long as the division doesn't involve a remainder. You often encounter natural numbers in real-life scenarios such as counting objects, steps, or events.

Set notation is a mathematical system used to define and describe a set, which is simply a collection of distinct objects or numbers. It allows us to specify a group of numbers or objects clearly and concisely. Sets are usually represented by curly braces \(\{ \}\) and the elements of sets are listed within them.

For instance, if we say \(\{1, 2, 3, 4, 5\}\), we are defining a set containing the numbers 1 through 5. We can also define sets using the builder or set-builder notation, where we provide a rule or condition for the elements. For example, in the set \(\{x \mid x \in \mathbf{N}\) and \(x \geq 100\}\), we are describing the set of natural numbers that are 100 or greater.

• The vertical bar \(|\) within the notation acts as a separator meaning "such that."
• The condition \(x \geq 100\) defines which members belong to this set.

Set notation is quite handy in mathematics for defining large or complex sets without listing all the elements individually.

Sets in mathematics can be classified as either finite or infinite, which describes the number of elements or members in a set.

A **finite set** is one that has a limited number of elements. For example, the set of people in a room is finite because you can count them, and there is a definite end. Finite sets have a specific, countable number of members.

• Example: \(\{2, 4, 6, 8, 10\}\) is a finite set with 5 elements.

An **infinite set**, on the other hand, is a set with endless or limitless elements. The set of natural numbers starting from 1 \(\{1, 2, 3, \ldots\}\) is infinite because it extends forever without stopping.

• Infinite sets cannot be "counted" in the usual sense since they go on indefinitely.

In the exercise provided, the set \(\{x \mid x \in \mathbf{N}\) and \(x \geq 100\}\) is infinite because there are no boundaries to the natural numbers starting from 100. The key difference is the existence of an endpoint. Finite sets have one, while infinite sets do not.