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= Subsets - Definition, Types, and Examples =

In set theory, a '''subset''' is a set whose elements all belong to another set. Subsets are fundamental in understanding the relationships between sets.

== Definition of Subset ==

A set <math>A</math> is called a '''subset''' of a set <math>B</math> if every element of <math>A</math> is also an element of <math>B</math>. This is written as:

<math>A \subseteq B</math>

This means:
<math>\forall x (x \in A \Rightarrow x \in B)</math>

If <math>A</math> is a subset of <math>B</math>, then all elements of <math>A</math> are contained in <math>B</math>.

== Types of Subsets ==

=== Proper Subset ===

If <math>A</math> is a subset of <math>B</math> and <math>A \neq B</math>, then <math>A</math> is called a '''proper subset''' of <math>B</math>, denoted by:

<math>A \subset B</math>

This means <math>A</math> contains some but not all elements of <math>B</math>.

=== Improper Subset ===

If <math>A = B</math>, then <math>A</math> is an '''improper subset''' of <math>B</math>. Every set is an improper subset of itself.

== Examples of Subsets ==

=== Example 1: ===
<math>A = \{1, 2\}</math> and <math>B = \{1, 2, 3, 4\}</math>  
Since all elements of <math>A</math> are in <math>B</math>, <math>A \subseteq B</math>.

=== Example 2: ===
<math>C = \{a, b\}</math> and <math>D = \{a, b\}</math>  
Here, <math>C = D</math>, so <math>C \subseteq D</math> and <math>D \subseteq C</math> (improper subsets).

=== Example 3: ===
<math>E = \emptyset</math> (the empty set) is a subset of every set, so <math>\emptyset \subseteq A</math> for any set <math>A</math>.

== Important Properties of Subsets ==

* '''Reflexivity''': Every set is a subset of itself, i.e., <math>A \subseteq A</math>.
* '''Transitivity''': If <math>A \subseteq B</math> and <math>B \subseteq C</math>, then <math>A \subseteq C</math>.
* The empty set is a subset of every set.
* A proper subset always has fewer elements than the original set.

== How to Check if a Set is a Subset ==

To verify if <math>A \subseteq B</math>:

1. Take each element of <math>A</math>.
2. Check if it is also an element of <math>B</math>.
3. If all elements of <math>A</math> belong to <math>B</math>, then <math>A</math> is a subset of <math>B</math>.

== Conclusion ==

Subsets describe the inclusion relationship between sets. Understanding subsets is essential for studying set operations, functions, and mathematical proofs. Recognizing proper and improper subsets helps clarify these relationships.

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