# Nonempty Set

A nonempty set is a set with at least one element. It is sometimes called a nonvoid set.

In math, a set is defined as a collection of distinct mathematical objects in which multiplicity matter doesn’t matter. What that means is that every unique element is counted exactly once, and a set is defined by its members. A set may be ordered or unordered.

We define a set S as nonempty if S ≠ ∅

If a nonempty set has exactly one element, it is called a singleton step.

## Importance of Nonempty Set

It is important in many proofs and equations that a set be nonempty, since computing with empty sets—just like doing arithmetic with 0—can lead to erratic or nonsensical results. Nonempty sets also form the basis of groups, a very important concept in set theory and topology.

## Examples of Nonempty Sets

Any grouping of elements which satisfies the properties of a set and which has at least one element is an example of a non-empty set, so there are many varied examples.

The set S= {1} with just one element is an example of a nonempty set. S so defined is also a singleton set.

The set S = {1,4,5} is a nonempty set.

The set of all real numbers is another example of a nonempty set. It contains an infinite number of elements, so it has more than one element and satisfies our definition.

## References

Weisstein, Eric W. “Nonempty Set.” From MathWorld–A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/NonemptySet.html on August 19, 2018.
Definition: Non-empty Set. From Proof-Wiki. Retrieved from https://proofwiki.org/wiki/Definition:Non-Empty_Set on August 23,2018.
Wang, Ka Lun. Teaching Notes 1: Sets. University of Hawaii Department of Math. Retrieved from https://math.hawaii.edu/~wongkl/Notes1.pdf on August 23, 2018.

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