# Set/Definition by Predicate/Examples

## Examples of Set Definition by Predicate

### 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:

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

### People who Love Romeo

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

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