Definition:Set Equivalence
Definition
Let $S$ and $T$ be sets.
Then $S$ and $T$ are equivalent if and only if:
That is, if and only if they have the same cardinality.
This can be written $S \sim T$.
If $S$ and $T$ are not equivalent we write $S \nsim T$.
Also known as
Other terms that are used that mean the same things as equivalent are:
- Equipotent (equalness of power), from which we refer to equivalent sets as having the same power
- Equipollent (equalness of strength)
- Equinumerous or equinumerable (equalness of number)
- Similar.
Also denoted as
Some sources use $S \simeq T$ or $S \approx T$ instead of $S \sim T$ to denote equivalence.
Other notations for $S \sim T$ include:
- $S \mathrel {\operatorname {Eq} } T$
- $\map {\mathrm {Eq} } {S, T}$
Examples
Arbitrary Example $1$
Let:
\(\ds S\) | \(=\) | \(\ds \set {2, 4, 6, 8}\) | ||||||||||||
\(\ds T\) | \(=\) | \(\ds \set {3, 5, 7, 9}\) |
There exists a bijection between $S$ and $T$, that is: $f : x \mapsto x + 1$, for example.
They both have the same cardinality, that is $4$.
Arbitrary Example $2$
Let:
\(\ds A\) | \(=\) | \(\ds \set {1, 2, 3}\) | ||||||||||||
\(\ds B\) | \(=\) | \(\ds \set {1, 2, 3, 4, 5}\) |
There does not exists a bijection between $A$ and $B$.
That is because they do not have the same cardinality.
Also see
- Results about set equivalence can be found here.
Historical Note
The notion of set equivalence was first introduced by Georg Cantor in $1878$.
The term he used was Mächtigkeit.
Sources
- 1915: Georg Cantor: Contributions to the Founding of the Theory of Transfinite Numbers ... (previous) ... (next): First Article: $\S 1$: The Conception of Power or Cardinal Number
- 1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 2$. Infinite sets
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 13$: Arithmetic
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.3$: Definition $1.9 \ \text{(c)}$
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.7$. Similar sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 17$: Finite Sets
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 2.3$: Equivalence of sets
- 1971: Patrick J. Murphy and Albert F. Kempf: The New Mathematics Made Simple (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets: Equivalent Sets: Definition: $1.5$
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.1$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 4$: Number systems $\text{I}$: A set-theoretic approach
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 8$. Equipotent sets; cardinal arithmetic; $\mathbf N$
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.4$: Functions
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $3$: Cardinality: Definition $1$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equinumerable (equipollent, equipotent)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equivalent (of sets)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): one-to-one correspondence ($\text {1-1}$ correspondence)
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $2$. Definition of Equivalence. The Concept of Cardinality. The Axiom of Choice: Definition $2.1$
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equinumerable (equipollent, equipotent)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equivalent (of sets)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): one-to-one correspondence (one-one or $\text {1-1}$ correspondence)