Sets

A rigorous foundation in sets, membership, subsets, equality, cardinality, and power sets.

Set operations select regions according to membership conditions. Two overlapping sets sit inside a universal set. Distinct shading marks intersection, union-only regions, and complement so logical words correspond to visible regions.

Precise definition

A set is a collection determined by membership: for each object, it is meaningful to ask whether it belongs. Sets ignore order and repetition, so $\{1,2,2\}=\{2,1\}$. Two sets are equal exactly when they have the same elements. The empty set $\varnothing$ has no elements and is a subset of every set.

Notation and mathematical language

Write $x\in A$ for membership and $A\subseteq B$ when every element of $A$ lies in $B$. A proper subset $A\subsetneq B$ is unequal to $B$. The cardinality $|A|$ counts elements for finite $A$. The power set $\mathcal P(A)$ is the set of all subsets of $A$.

Conceptual picture

A set can be specified by a roster or a membership rule. A power set is a collection whose elements are themselves sets; the empty subset and the entire original set are always included. A set of size $n$ has $2^n$ subsets because each element independently has an in/out choice.

Fully worked example

Interpretation and application

Sets define solution spaces, sample spaces, categories, database results, and mathematical structures. Real categories can overlap or have uncertain membership; the mathematical set model is exact only after the classification rule is stated.