Definition:Count

Definition

To count a set $S$ is to establish a bijection between $S$ and an initial segment $\N_n$ of the natural numbers $\N$.

If $S \sim \N_n$ (where $\sim$ denotes set equivalence) then we have counted $S$ and found it has $n$ elements.

If $S \sim \N$ then $S$ is infinite but countable.

Also known as

The process of counting is also known as enumeration, although that has a more specialized meaning on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see

• Definition:Number of Elements
• Cardinality of Finite Set is Well-Defined
• Definition:Uncountable Set

• Principle of Counting

Sources

• 1939: E.G. Phillips: A Course of Analysis (2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Number: $1.1$ Introduction
• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 2.4$: Definition $2.4$
• 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
• 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Discrete and Continuous Variables
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): combinatorics
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): combinatorics
• 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 $\text{A}.5$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): count