Subsets

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 $A$ is called a subset of a set $B$ if every element of $A$ is also an element of $B$. This is written as:

$A\subseteq B$

This means: $\mathrm{\forall }x(x\in A\Rightarrow x\in B)$

If $A$ is a subset of $B$, then all elements of $A$ are contained in $B$.

Types of Subsets

Proper Subset

If $A$ is a subset of $B$ and $A\ne B$, then $A$ is called a proper subset of $B$, denoted by:

$A\subset B$

This means $A$ contains some but not all elements of $B$.

Improper Subset

If $A=B$, then $A$ is an improper subset of $B$. Every set is an improper subset of itself.

Examples of Subsets

Example 1:

$A=\{1,2\}$ and $B=\{1,2,3,4\}$ Since all elements of $A$ are in $B$, $A\subseteq B$.

Example 2:

$C=\{a,b\}$ and $D=\{a,b\}$ Here, $C=D$, so $C\subseteq D$ and $D\subseteq C$ (improper subsets).

Example 3:

$E=\mathrm{\varnothing }$ (the empty set) is a subset of every set, so $\mathrm{\varnothing }\subseteq A$ for any set $A$.

Important Properties of Subsets

• Reflexivity: Every set is a subset of itself, i.e., $A\subseteq A$.
• Transitivity: If $A\subseteq B$ and $B\subseteq C$, then $A\subseteq C$.
• 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 $A\subseteq B$:

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

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.