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Nonempty Set

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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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