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.