# Definition:Set/Definition by Predicate

## Definition

An object can be specified by means of a **predicate**, that is, in terms of a property (or properties) that it possesses.

Whether an object $x$ possesses a particular property $P$ is either true or false (in Aristotelian logic) and so can be the subject of a propositional function $\map P x$.

Hence a set can be specified by means of such a propositional function:

- $S = \set {x: \map P x}$

which means:

or, more formally:

In this context, we see that the symbol $:$ is interpreted as **such that**.

## Also known as

This construction is sometimes known as the **set-builder notation** or as **set comprehension**.

This is also sometimes rendered as **set builder notation**.

An alternative notation for this is $S = \set {x \mid \map P x}$, but it can be argued that the use of $\mid$ for **such that** can cause ambiguity and confusion, as $\mid$ has several other meanings in mathematics.

On the other hand, if the expression defining the predicate is thick with $:$ characters, it may improve clarity to use $\mid$ for **such that** after all.

Some authors, mindful of such confusion, use the notation $S = \set {x; \map P x}$ as the semicolon is relatively rare in mathematical notation.

Sometimes it is convenient to abbreviate the notation by simply writing $S = \set {\map P x}$ or even just $S = \set P$.

For example, to describe the set $\set {x \in \R: \map f x \le \map g x}$ (for appropriate functions $f, g$), one could simply use $\set {f \le g}$.

## Axiomatic Set Theory

In the context of axiomatic set theory, a more strictly rigorous presentation of this concept is:

- $S = \set {x \in A: \map P x}$

which means:

or, more formally:

This presupposes that all the objects under consideration for inclusion in $S$ *already* belong to some previously-defined set $A$.

Thus any set $S$ can be expressed as:

- $S = \set {s: s \in S}$

See the Axiom of Specification.

## Examples

### University Professors

An example in natural language of a set definition by predicate is:

- $S := \text {the set of all university professors}$

### Set of Integers $x$ such that $2 \le x$

Let $S$ be the set defined as:

- $S := \set {x \in \Z: 2 \le x}$

Then $S$ is the set of all integers greater than or equal to $2$:

- $S = \set {2, 3, 4, \ldots}$

### Set of Integers $x$ such that $x \le 5$

Let $S$ be the set defined as:

- $S := \set {x \in \Z: x \le 5}$

Then $S$ is the set of all integers less than or equal to $5$:

- $S = \set {\ldots, 2, 3, 4, 5}$

### Set Indexed by Natural Numbers between $1$ and $100$

Let $V$ be the set defined as:

- $V := \set {v_i: 1 \le i \le 100, i \in \N}$

Then $V$ is the set of the $100$ elements:

- $V = \set {v_1, v_2, \ldots, v_{100} }$

and can also be written:

- $V := \set {v_i: i = 1, 2, \ldots, 100}$

or even:

- $V := \set {v_i: 1 \le i \le 100}$

as it is understood that the domain of $i$ is the set of natural numbers.

### Set Indexed by Natural Numbers between $1$ and $10$

Let $U$ be the set defined as:

- $U := \set {u_i: 1 < i < 10, i \in \N}$

Then $U$ has exactly $8$ elements:

- $U = \set {u_2, u_3, u_4, u_5, u_6, u_7, u_8, u_9}$

## Also see

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Sets - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 2$: The Axiom of Specification - 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 1.6$: Functions - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.1$ - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.1$: Sets - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.1$. Sets - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Introduction: Set-Theoretic Notation - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 4$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S1.1$: Sets and Membership - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Sets - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$ - 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: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 3$: Statements and conditions; quantifiers - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 1.2$: Sets - 1983: George F. Simmons:
*Introduction to Topology and Modern Analysis*... (previous) ... (next): $\S 1$: Sets and Set Inclusion - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.3$: Notation for Sets - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.1$: Sets - 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $1$. Notation, Conventions: $6$ - 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: Sets and Subsets - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): $\S 1.2$: Elements, my dear Watson - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): Appendix $\text{A}.1$