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In mathematics, natural numbers are the set of positive integers starting from 1, 2, 3, and so on. These numbers are the ones we naturally use for counting objects in everyday life. Natural numbers do not include zero, fractions, or negative numbers.

For example, if you have 5 apples, you count them using natural numbers: 1 apple, 2 apples, 3 apples, etc. Natural numbers are symbolized by the letter \textbf{N} and can be written as:\[ \mathbb{N} = \{1, 2, 3, 4, 5, \ldots\} \]

Natural numbers are the building blocks of arithmetic. They appear in almost every branch of mathematics and play a crucial role in number theory.

Odd numbers are a specific subset of natural numbers. An odd number is an integer which is not divisible by 2. When you divide an odd number by 2, you will always get a remainder of 1. If you look at any sequence of numbers, odd numbers are the ones that alternate with even numbers.

Examples of odd numbers include 1, 3, 5, 7, 9, and so on. Mathematically, we can express odd numbers as:\[ 2n + 1 \text { where } n \text { is an integer} \]

In the set provided in the original exercise, we had to filter out these odd numbers from the list of natural numbers less than 14. The resulting set of odd numbers is: \{1, 3, 5, 7, 9, 11, 13\}.

Set notation is a mathematical way to describe a collection of objects, known as elements. Sets are typically written using curly braces \(\{\} \), with the elements of the set listed within them. For example, the set of natural numbers less than 14 is written as \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}.

In the original exercise, the set was defined using set-builder notation, which describes the properties of the elements of the set rather than listing every element individually. The set-builder notation used was: \[ \{x \mid x \text { is an odd natural number less than } 14\} \]

This means we are looking for all odd natural numbers less than 14. Set notation is fundamental in math because it provides a compact and precise way to represent and work with collections of objects.