Types of Sets

Why This Matters

In set theory, understanding the different types of sets isn't just about memorizing definitions—it's about recognizing the structural properties that make sets behave in fundamentally different ways. You're being tested on your ability to classify sets by their cardinality, relationships to other sets, and topological properties. These distinctions form the foundation for nearly everything else in mathematics, from functions and relations to probability and analysis.

When you encounter exam questions, they'll often ask you to identify which type of set fits a given description or to explain why certain set properties matter. Don't just memorize that the empty set has no elements—understand why it's a subset of every set and how that fact gets used in proofs. The goal is to see each set type as an example of a deeper principle at work.

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Sets Defined by Size (Cardinality)

The most fundamental way to classify sets is by how many elements they contain. Cardinality tells us whether a set is empty, finite, or infinite—and if infinite, whether it's countable or uncountable.

Empty Set

• Contains no elements—denoted by $\emptyset$ or $\{\}$, with cardinality zero
• Subset of every set—this follows logically since there's no element in $\emptyset$ that could fail to be in another set
• Unique in set theory—there is exactly one empty set, making it foundational for building other mathematical structures

Singleton Set

• Contains exactly one element—written as $\{a\}$ where $a$ is the sole member
• Cardinality of one—the simplest non-empty set possible
• Not the same as its element—the set $\{a\}$ is distinct from the element $a$ itself, a crucial distinction in formal set theory

Finite Set

• Has a countable, limited number of elements—you can list all members and eventually stop
• Cardinality is a natural number—for example, $\{1, 2, 3\}$ has cardinality 3
• Closed under many operations—unions and intersections of finite sets remain finite

Infinite Set

• Has unlimited elements—no natural number can express its cardinality
• Comes in different sizes—not all infinite sets are equally large (countable vs. uncountable)
• Examples include $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{R}$—the natural numbers, integers, and real numbers respectively

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Sets Defined by Countability

Within infinite sets, mathematicians distinguish between those that can be listed in sequence and those that cannot. This distinction, formalized by Cantor, reveals that infinities come in different magnitudes.

Countable Sets

• Elements correspond one-to-one with natural numbers—you can "count" them even if the counting never ends
• Includes all finite sets plus some infinite sets—$\mathbb{N}$ and $\mathbb{Z}$ are both countable
• Key technique: bijection with $\mathbb{N}$—proving countability means constructing this correspondence explicitly

Uncountable Sets

• Cannot be matched one-to-one with natural numbers—there are "too many" elements
• The real numbers $\mathbb{R}$ are the classic example—Cantor's diagonal argument proves this
• Strictly larger cardinality than countable sets—uncountable sets represent a higher order of infinity

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Sets Defined by Relationships

Sets don't exist in isolation—they relate to each other through inclusion, equality, and equivalence. These relationships are essential for understanding set operations and logical arguments.

Universal Set

• Contains all elements under consideration—denoted $U$, it defines the "universe" for a given problem
• Every other set is a subset of $U$—complements and other operations are defined relative to it
• Context-dependent—the universal set changes based on what you're studying (integers in number theory, points in geometry)

Subset

• Set $A$ is a subset of $B$ if every element of $A$ is in $B$—written $A \subseteq B$
• Includes the possibility of equality—$A \subseteq A$ is always true
• The empty set is a subset of every set—a critical fact used in many proofs

Proper Subset

• $A$ is a proper subset of $B$ if $A \subseteq B$ and $A \neq B$—written $A \subset B$
• $B$ must contain at least one element not in $A$—proper subsets are strictly "smaller"
• Used to establish strict hierarchies—important when proving one set is genuinely contained in another

Power Set

• The set of all subsets of $A$—denoted $\mathcal{P}(A)$ or $2^A$
• If $|A| = n$, then $|\mathcal{P}(A)| = 2^n$—this exponential growth is a key formula
• Always includes $\emptyset$ and $A$ itself—these are the "extreme" subsets

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Sets Defined by Element Comparison

When comparing two sets, we ask: Do they share elements? Do they have the same elements? Do they have the same number of elements? Each question leads to a different classification.

Equal Sets

• Contain exactly the same elements—written $A = B$
• Order and repetition don't matter—$\{1, 2, 3\} = \{3, 1, 2\}$
• Proven by showing mutual subset inclusion—$A = B$ iff $A \subseteq B$ and $B \subseteq A$

Equivalent Sets

• Have the same cardinality—written $|A| = |B|$
• Elements can be completely different—$\{1, 2\}$ and $\{a, b\}$ are equivalent
• Established by bijection—a one-to-one correspondence between elements proves equivalence

Disjoint Sets

• Share no common elements—their intersection is empty: $A \cap B = \emptyset$
• Can still have the same cardinality—$\{1, 2\}$ and $\{3, 4\}$ are disjoint but equivalent
• Foundation for partitions—a set can be divided into pairwise disjoint subsets

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Sets Defined by Topological Properties

In analysis and topology, sets are classified by their boundary behavior. These properties become essential when studying continuity, limits, and convergence.

Open Sets

• Contains none of its boundary points—every point has a neighborhood entirely within the set
• The interval $(a, b)$ is the classic example—endpoints $a$ and $b$ are excluded
• Fundamental to defining continuity—a function is continuous iff preimages of open sets are open

Closed Sets

• Contains all of its boundary points—includes its "edge" elements
• The interval $[a, b]$ is the classic example—endpoints $a$ and $b$ are included
• Complement of an open set is closed—and vice versa, a key duality in topology

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Quick Reference Table

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Self-Check Questions

1. What property do the empty set and singleton set share regarding their relationship to other sets, and how do their cardinalities differ?

2. Given two infinite sets, how would you determine whether they are countable or uncountable? What proof technique distinguishes them?

3. Compare and contrast equal sets and equivalent sets. Can two sets be equivalent without being equal? Can they be equal without being equivalent?

4. If $A = \{1, 2, 3\}$, how many elements does $\mathcal{P}(A)$ contain? List three subsets of $A$ that would appear in the power set.

5. Explain why the interval $(0, 1)$ is open while $[0, 1]$ is closed. What would you call a set that is both open and closed, and can you give an example?