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# empty set

An empty set is a set $\emptyset$ that contains no elements. The Zermelo-Fraenkel Axioms of set theory imply that there exists an empty set. One constructs an empty set by starting with any set $X$ and then applying the axiom of separation to form the empty set $\emptyset:=\{x\in X\mid x\neq x\}$.

An empty set is a subset of every other set, and any two empty sets are equal. Alternative notations for the empty set include $\{\}$ and $\varnothing$.

Synonym:

null set

Type of Math Object:

Definition

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Reference

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## Mathematics Subject Classification

03-00*no label found*65H05

*no label found*65H10

*no label found*

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## Comments

## empty set

Can you explain me why an empty set is a subset of every other set?

By definition, or is there a demonstration, or it's a convention?

And since any two empty sets are equal, do you allow me to say that every man that is also a woman is a subset of irrational numbers?

## Re: empty set

A is a subset of B, if any any element of A is also an element of B.

If A = {}, it fulfils the above requirement; since A is empty, the requirement is empty, we require nothing! (Try to find some element of {} not belonging to B !)

Jussi

## inclusion of empty set

What I want to point out is that you must explicitly mention when empty set is an element of a set. The set of my customers is sometimes big but, unfortunately, it's happen to be empty occasionally. But the set of my actions in my office can't be empty, even if I fall asleep sometimes: at least, I get older every second.