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= Proper Subset - Definition and Examples =

A '''proper subset''' is a special kind of subset where all elements of one set are contained in another set, but the two sets are not equal. In other words, the proper subset must have fewer elements than the original set.

== Definition of Proper Subset ==

A set <math>A</math> is called a '''proper subset''' of a set <math>B</math> if:

* Every element of <math>A</math> is in <math>B</math>, and
* <math>A</math> is not equal to <math>B</math>.

This is denoted by:

<math>A \subset B</math>

Mathematically:

<math>A \subset B \iff (A \subseteq B) \wedge (A \neq B)</math>

where <math>\subseteq</math> denotes "subset or equal".

== Examples of Proper Subsets ==

=== Example 1: ===  
<math>A = \{1, 2\}</math> and <math>B = \{1, 2, 3\}</math>  
Since all elements of <math>A</math> are in <math>B</math> and <math>A \neq B</math>,  
<math>A \subset B</math> (A is a proper subset of B).

=== Example 2: ===  
<math>C = \{\text{apple}, \text{banana}\}</math> and <math>D = \{\text{apple}, \text{banana}, \text{cherry}\}</math>  
Here, <math>C \subset D</math>.

=== Example 3: ===  
<math>\emptyset</math> (the empty set) is a proper subset of every non-empty set. For example,  
<math>\emptyset \subset \{1\}</math>.

== Important Notes ==

* A proper subset cannot be equal to the original set.
* If <math>A = B</math>, then <math>A</math> is a subset but '''not''' a proper subset of <math>B</math>.
* The empty set is a proper subset of any set except itself.

== Difference Between Subset and Proper Subset ==

{| class="wikitable"
! Property !! Subset <math>(\subseteq)</math> !! Proper Subset <math>(\subset)</math>
|-
| Definition || All elements of A are in B, A may be equal to B || All elements of A are in B, but A is not equal to B
|-
| Symbol || <math>A \subseteq B</math> || <math>A \subset B</math>
|-
| Example || <math>\{1, 2, 3\} \subseteq \{1, 2, 3\}</math> (equal sets) || <math>\{1, 2\} \subset \{1, 2, 3\}</math> (proper subset)
|}

== Conclusion ==

The concept of '''proper subsets''' is essential for understanding detailed relationships between sets. Proper subsets help distinguish when one set is strictly contained within another without being equal.

[[Category:Set Theory]]
[[Category:Mathematics]]
[[Category:Types of Sets]]