Set Definition by Predicate/Examples

Examples of Set Definition by Predicate

Men

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

$S := \set {x: x \text { is a man} }$

Thus $S$ is the set which contains all men and only men.

University Professors

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

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

Musical Mathematicians

Let $M$ denote the set of all the mathematicians in the world.

Let $I$ denote the set of all people who can play a musical instrument.

Let $S$ denote the set of all mathematicians who can play a musical instrument.

Then we can define $S$ as:

$S := \set {x: x \in M \text { and } x \in I}$

or as:

$S := \set {x \in M: x \in I}$

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:

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

Even Integers

Let $S$ be the set defined as:

$S := \set {i: \text {$i$ is an integer and there exists an integer $j$ such that $i = 2 j$} }$

Then $S$ is the set of all even integers.

People who Love Romeo

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

$D := \set {y: \text {$y$ loves Romeo} }$

Cube Numbers

Let $C$ be the set defined as:

$C := \set {n: n = k^3 \text { and } k = 1, 2, \dotsc}$

Then $C$ is the set of all cube numbers.

Sums of Two Squares

Let $S$ be the set defined as:

$S := \set {n: n = x^2 + y^2 \text { and } x, y \in \Z}$

The $C$ is the set of all integers which can be written as the sum of two squares.

United States Senate

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

$S := \text {the members of the United States Senate}$