Understanding Subsets and Power Sets in Math

A finite set and its subsets form foundational elements of set theory, with particular importance in formulating combinatorial and algebraic arguments in JEE-level mathematics. Binary choices for element inclusion lead systematically to the concept of the power set.

Notation and Formal Definition of Subsets and Power Sets

Definition: Let $A$ and $B$ be sets. $A$ is termed a subset of $B$ if every element of $A$ is also an element of $B$, which is denoted as $A\subseteq B$. If $A \subseteq B$ and $A \neq B$, then $A$ is called a proper subset of $B$, denoted $A \subset B$.

The empty set, denoted $\varnothing$ or $\{\}$, is a subset of every set. The set $A$ is always an improper subset of itself. The collection of all subsets of a set $S$ is called the power set of $S$ and is denoted $\mathcal{P}(S)$.

If $S$ is a set with $n$ elements, then $|\mathcal{P}(S)| = 2^n$ distinct subsets exist. For $S = \{x, y, z\}$, $\mathcal{P}(S) = \{\varnothing, \{x\}, \{y\}, \{z\}, \{x, y\}, \{y, z\}, \{z, x\}, \{x, y, z\}\}$.

For foundational relations among sets, see Sets, Relations, And Functions.

Classification of Subsets and Cardinality Results

Subsets are classified by inclusion. A proper subset omits at least one element of the parent set, whereas the improper subset is the set itself. The notation $\subset$ denotes proper subset, whereas $\subseteq$ denotes subset (proper or improper).

If $A$ is a set with $n$ elements, the total number of subsets is $2^n$, and the number of proper subsets is $2^n - 1$, since the proper subset count excludes $A$ itself.

For categorical comparisons on sets and their subsets, refer to Types Of Sets.

Theorems Involving Subsets and Power Sets

Let $A$, $B$, and $C$ be sets.

Result: If $A\subseteq B$ and $B \subseteq C$, then $A \subseteq C$. This follows since membership passes transitively: if $x \in A$, then $x \in B$, and hence $x \in C$.

Result: For any set $A$, we have $\varnothing \subseteq A$ and $A \subseteq A$. The condition is immediate: an empty set contains no elements, so its subset relation is always satisfied.

Result: For $|A|=n$, $|\mathcal{P}(A)|=2^n$. Each element either belongs or does not belong to a subset; thus, by the multiplication principle, $2^n$ subsets. For further exploration, see Set Theory Concepts.

Symbolic Representation and Key Formulae

• Subset: $\subseteq$
• Proper subset: $\subset$
• Power set: $\mathcal{P}(S)$
• Cardinality of power set: $2^n$

Differentiating Subsets and Power Sets

Worked Examples: Evaluation of Subsets and Power Sets

Example: For $A = \{a, b, c, d\}$, to find the number of subsets and proper subsets.

Solution: The set $A$ has $4$ elements ($n=4$). Therefore, number of subsets is $2^4 = 16$.

The number of proper subsets is $2^4 - 1 = 15$.

For expanded examples, see Union, Intersection, And Difference Of Sets.

Example: For $S = \{1, 2\}$, write all subsets and the power set.

Solution: The subsets are $\varnothing$, $\{1\}$, $\{2\}$, $\{1,2\}$. Thus, the power set is $\mathcal{P}(S) = \{\varnothing, \{1\}, \{2\}, \{1,2\}\}$.

Example: If $B = \{x, y, z\}$ and $A = \{x, y\}$, verify the subset relation.

Solution: All elements of $A$ are in $B$, but $z \in B$ and $z \notin A$. Thus, $A \subset B$.

Example: For $C = \varnothing$, list $\mathcal{P}(C)$.

Solution: The empty set has exactly one subset, itself. Thus, $\mathcal{P}(\varnothing) = \{\varnothing\}$.

Further practice is available at Venn Diagram In Set Theory.

Common Misunderstandings in Subset and Power Set Concepts

• Assuming the null set is not a subset of every set
• Confusing the number of subsets ($2^n$) with that of proper subsets ($2^n - 1$)
• Incorrectly including elements outside the original set in subsets
• Interpreting the power set as a single large set, rather than a set of sets
• Misusing symbols $\subset$, $\subseteq$

Refer to Subsets And Power Sets for detailed sub-concept clarifications.