# What is a non trivial set?

A non-trivial set refers to a subset of a larger set that is neither empty nor contains all the elements of the original set. In other words, it is a subset that has at least one element but is not the entirety of the set.

To understand the concept of a non-trivial set, let’s consider an example. Imagine you have a set of fruits, which includes apples, oranges, and bananas. Now, if you take a subset of this set that includes only apples, it would be a non-trivial set because it is not empty and does not contain all the fruits in the original set.

On the other hand, if you take a subset that includes all the fruits in the original set, it would not be a non-trivial set. Similarly, if you take a subset that does not include any fruit, it would also not be a non-trivial set as it would be an empty set.

The concept of non-trivial sets is important in various mathematical fields, such as set theory, algebra, and topology. It allows mathematicians to distinguish between subsets that have some elements and those that do not.

Non-trivial sets can have different sizes, ranging from containing just one element to having several elements. For example, in the set of even numbers, the subset consisting of only the number 2 would be a non-trivial set. Similarly, a subset containing the numbers 2, 4, and 6 would also be a non-trivial set.

In my personal experience as a mathematician, non-trivial sets often arise when dealing with problems that involve partitioning or classifying elements of a larger set. They provide a way to focus on specific elements or subsets of interest without considering the entire set.

To summarize, a non-trivial set is a subset of a larger set that contains at least one element but is not the entirety of the set. It plays a crucial role in various mathematical fields and allows for the distinction between subsets that have some elements and those that do not.