Subset and Proper Subset Relations in Set Theory
In discrete mathematics and formal logic, understanding set inclusion is foundational. A set $A$ is related to another set $B$ based on whether all elements of $A$ are contained within $B$.
Definitions and Mathematical Symbols
- 1Subset ($A \subseteq B$): Every element in $A$ is also in $B$:
\forall x \in A \implies x \in B
Every set is trivially a subset of itself: $A \subseteq A$.
- 1Proper Subset ($A \subset B$): $A$ is a subset of $B$, but $A \ne B$. That is, $B$ contains at least one element not present in $A$:
A \subseteq B \quad \text{and} \quad A \ne B
- 1Empty Set ($\emptyset$): The empty set is a subset of every set: $\emptyset \subseteq B$ for all $B$.
Subset vs Proper Subset Comparison Table
| Relation | Symbol | Can $A = B$? | Example |
|---|---|---|---|
| Subset | $A \subseteq B$ | Yes | $\{1, 2\} \subseteq \{1, 2\}$ |
| Proper Subset | $A \subset B$ | No (strictly smaller) | $\{1, 2\} \subset \{1, 2, 3\}$ |
| Not a Subset | $A \not\subseteq B$ | N/A | $\{1, 4\} \not\subseteq \{1, 2, 3\}$ |
| Set Equality | $A = B$ | Always ($A \subseteq B$ and $B \subseteq A$) | $\{1, 2\} = \{2, 1\}$ |