# Subset/Examples

## Examples of Subsets

### British People are Subset of People

Let $B$ denote the set of British people.

Let $P$ denote the set of people.

Then $B$ is a proper subset of $P$:

- $B \subsetneq P$

### Subset of Alphabet

Let $S$ denote the capital letters of the (English) alphabet:

- $S = \set {A, B, C, D, \dotsc, Z}$

Then $\set {A, B, C}$ is a subset of $S$:

- $\set {A, B, C} \subseteq S$

### Integers are Subset of Real Numbers

The set of integers $\Z$ is a proper subset of the set of real numbers $\R$:

- $\Z \subsetneq \R$

### Initial Segment is Subset of Integers

The (one-based) initial segment of the natural numbers $\N^*_{\le n}$ of integers $\Z$ is a proper subset of the set of integers $\Z$:

- $\N^*_{\le n} \subsetneq \Z$

### Even Numbers form Subset of Integers

The set of even integers forms a subset of the set of integers $\Z$.

### Even Numbers form Subset of Real Numbers

The set of even integers forms a subset of the set of real numbers $\R$.

### Abstract Example $1$

Let $P$ denote the set:

- $P = \set {1, 2, 3, 4}$

Let $Q$ denote the set:

- $Q = \set {2, 4}$

Then:

- $Q \subseteq P$

but note that:

- $Q \notin P$

### Abstract Example $2$

Let $A$ denote the set:

- $A := \set {\dfrac 1 2, 1, \sqrt 2, e, \pi}$

Let $C$ denote the set:

- $C := \set {x \in \R: x > 0}$

Then:

- $A \subseteq C$

### Abstract Example $3$

Let $A$ denote the set:

- $A := \set {1, 2, 3}$

Let $B$ denote the set:

- $B := \set {\&, 3, +, 1, 2}$

Then:

- $A \subseteq B$
- $A \subseteq A$
- $B \subseteq B$

### Romeo and Juliet

Let $B$ denote the set:

- $B := \set {\textrm {Romeo}, \textrm {Juliet} }$

Let $D$ denote the set:

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

Then:

- $B \subseteq D$

presuming, of course, that Romeo actually loves himself, which may be doubted considering he took his own life.