Definition:Empty Set - ProofWiki

Contents

• 1 Definition
• 2 Axiomatic Set Theory
  • 2.1 Set Theory
  • 2.2 Class Theory
• 3 Notation
• 4 Existence of Empty Set
• 5 Examples
  • 5.1 Real Roots of $x^2 + 1 = 0$
• 6 Also known as
• 7 Also see
• 8 Historical Note
• 9 Linguistic Note
• 10 Technical Note
• 11 Sources

Definition

The empty set is a set which has no elements.

That is, $x \in \O$ is false, whatever $x$ is.

It is usually denoted by some variant of a zero with a line through it, for example $\O$ or $\emptyset$, and can always be represented as $\set {}$.

Axiomatic Set Theory

The concept of the empty set is axiomatised in the Axiom of the Empty Set in Zermelo-Fraenkel set theory:

The Axiom of the Empty Set posits the existence of a set which has no elements.

Depending on whether this axiom is declared in the context of set theory or class theory, it exists in different forms.

Set Theory

$\exists x: \forall y: \paren {\neg \paren {y \in x} }$

Class Theory

In class theory, the existence of the empty class is not axiomatic, as it has been derived from previous axioms.

Hence the Axiom of the Empty Set takes this form:

The empty class $\O$ is a set, that is:

$\O \in V$

where $V$ denotes the basic universe.

Notation

The symbols $\O$ and $\emptyset$ used for the empty set are properly considered as stylings of $0$ (zero), and not variants of the Greek Phi: $\Phi, \phi, \varphi$.

Despite this, some sources call the symbol phi (pronounced fie).

Some sources maintain that it is a variant on the Norwegian / Danish / Faeroese letter Ø.

The symbol $\O$ as presented here is a relatively new invention. Books prior to approximately $1960$ or $1970$ tend to use something less distinctive:

Some such sources use $\Box$ as the symbol for the the empty set, but this is rare. Other sources use $0$ (that is, the zero digit). Yet others use $O$ (the capital letter).

None of these are recommended.

The preferred symbol on $\mathsf{Pr} \infty \mathsf{fWiki}$ is $\O$ for its completely unambiguous interpretation and aesthetically pleasing, clean presentation.

Also note that $\set {}$ can always be used.

Existence of Empty Set

Some authors have problems with the existence (or not) of the empty set:

• 1965: J.A. Green: Sets and Groups: $\S 1.3$:

If $A, B$ are disjoint, then $A \cap B$ is not really defined, because it has no elements. For this reason we introduce a conventional empty set, denoted $\O$, to be thought of as a 'set with no elements'. Of course this is a set only by courtesy, but it is convenient to allow $\O$ the status of a set.

• 1968: Ian D. Macdonald: The Theory of Groups: Appendix:

The best attitude towards the empty set $\O$ is, perhaps, to regard it as an interesting curiosity, a convenient fiction. To say that $x \in \O$ simply means that $x$ does not exist. Note that it is conveniently agreed that $\O$ is a subset of every set, for elements of $\O$ are supposed to possess every property.

• 2000: James R. Munkres: Topology (2nd ed.): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts

Now some students are bothered with the notion of an "empty set". "How", they say, "can you have a set with nothing in it?" ... The empty set is only a convention, and mathematics could very well get along without it. But it is a very convenient convention, for it saves us a good deal of awkwardness in stating theorems and proving them.

Such a philosophical position is considered by many mathematicians to be a timid attitude harking back to the mediaeval distrust of zero.

In any case, its convenience cannot be doubted:

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts: Introduction $\S 1$: Operations on Sets:

One may regard [the vacuous set] as a zero element that is adjoined to the collection of "real" subsets.

• 1965: Seth Warner: Modern Algebra: $\S 1$:

One practical advantage in admitting $\O$ as a set is that we may wish to talk about a set without knowing a priori whether it has any members.

• 1975: T.S. Blyth: Set Theory and Abstract Algebra: $\S 1$:

The courtesy of regarding this as a set has several advantages ... In allowing $\O$ the status of a set, we gain the advantage of being able to talk about a set without knowing at the outset whether or not it has any elements.

Some other sources allow the definition of the empty set, but because of the way natural numbers are defined, determine that it is neither finite nor infinite.

Examples

Real Roots of $x^2 + 1 = 0$

The set:

$S = \set {x \in \R: x^2 + 1 = 0}$

is an instance of a specification of the empty set.

Also known as

The empty set is sometimes called the null set, but this name is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$ because there is another concept for null set which ought not to be confused with this.

Some sources call the empty set the vacuous set.

Others call it the void set.

Also see

• Empty Set is Unique for a proof that it is justifiable to refer to $\O$ as the empty set.

• Definition:Non-Empty Set, a common phrasing used to denote any set but the empty set.
• Definition:Universal Set

• Results about the empty set can be found here.

Historical Note

The concept of the empty set was stated by Leibniz in his initial conception of symbolic logic.

The use of $\O$ has relevance to the computing industry, when $\emptyset$ is usually used in the rendering of source code to mean "zero", in order to distinguish it from $\mathrm O$ (the letter O).

In the same context, the letter O was sometimes seen rendered as $\odot$, so as to ensure its being differentiated from "zero".

The latter has fallen out of use, but it is still common for mathematicians, when writing their mathematics by hand, to strike through their zeroes out of habit.

Linguistic Note

The word vacuous literally means empty.

It derives from the Latin word vacuum, meaning empty space.

Technical Note

The $\LaTeX$ code for \(\O\) is \O .

The same symbol is also generated by \varnothing or \empty, but these are more unwieldy, and \O is preferred.

Sources

• 1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets
• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 1$: Operations on Sets
• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection
• 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 8$. Notations and definitions of set theory
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 3$: Unordered Pairs
• 1961: John G. Hocking and Gail S. Young: Topology ... (previous) ... (next): A Note on Set-Theoretic Concepts
• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Chapter $1$: A Common Language: $\S 1.1$ Sets
• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction: $1.3$. Notation
• 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.1$: Sets
• 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Set Theory: $1.2$: Sets and subsets
• 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.3$. Intersection
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
• 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Introduction: Special Symbols
• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $1$ Set Theory: $1$. Sets and Functions: $1.1$: Basic definitions
• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
• 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 3$
• 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): Chapter $1$: Sets, Functions, and Relations: $\S 1$: Sets and Membership
• 1971: Patrick J. Murphy and Albert F. Kempf: The New Mathematics Made Simple (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets: The Empty Set: Definition $1.2$
• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.16$
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Sets
• 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Sets
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.1$: Set Notation
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.1$: Sets and Subsets
• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 6$: Subsets
• 1981: G. de Barra: Measure Theory and Integration ... (previous) ... (next): Chapter $1$: Preliminaries: $1.1$ Set Theory
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $14.$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets
• 1989: George S. Boolos and Richard C. Jeffrey: Computability and Logic (3rd ed.) ... (previous) ... (next): $1$ Enumerability
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): empty: 1.
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): null set: 1.
• 1991: Richard S. Millman and George D. Parker: Geometry: A Metric Approach with Models (2nd ed.) ... (previous) ... (next): $\S 1.2$: Sets and Equivalence Relations
• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.2$: Operations on Sets
• 1993: Richard J. Trudeau: Introduction to Graph Theory ... (previous) ... (next): $2$. Graphs: Sets: Definition $2$
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): empty set
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): null set (empty set)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): set (class)
• 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $4$
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Unordered Pairs and their Relatives
• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson: Example $1.2.1$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): empty set
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): null set (empty set)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): set (class)
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 1$ What is infinity?
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.1$: Definition $\text{A}.1$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): empty set
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): empty set