Key Set Theory Notation

Why This Matters

Set theory notation is the foundation of mathematical logic—it's the language you'll use to express everything from basic membership claims to complex proofs. When you encounter problems involving relations, functions, probability, or formal reasoning, you're working with sets whether you realize it or not. The notation covered here appears constantly in exam questions, and understanding it fluently means you can parse problem statements quickly and express your reasoning precisely.

But here's what really matters for your exam: you're being tested on your ability to use these symbols correctly in context, not just recognize them. Can you translate between set-builder notation and roster notation? Do you know when to use $\subseteq$ versus $\subset$? Can you compute a Cartesian product or power set without hesitation? Don't just memorize symbols—know what operation or relationship each one represents and when to apply it.

---

Membership and Basic Set Notation

These symbols establish the fundamental vocabulary of set theory—how we define sets and describe what's in them. Master these first, as every other concept builds on membership relationships.

Set Notation: { }

• Curly braces define a set—the most basic way to collect distinct objects into a single mathematical entity
• Order is irrelevant and duplicates collapse—$\{1, 2, 3\} = \{3, 1, 2\} = \{1, 1, 2, 3\}$, a key distinction from sequences or lists
• Roster notation lists elements explicitly, making it ideal for finite sets with enumerable members

Element Of: ∈

• Indicates membership in a set—if $A = \{1, 2, 3\}$, then $2 \in A$ reads "2 is an element of A"
• Connects individual objects to collections—this is the fundamental relationship in all of set theory
• Appears in quantified statements—you'll see $x \in S$ constantly in definitions, theorems, and proof structures

Not an Element Of: ∉

• Negates membership—if $A = \{1, 2, 3\}$, then $4 \notin A$ states that 4 is not in the set
• Essential for proofs by contradiction—showing $x \notin A$ often requires demonstrating that $x$ fails some defining property
• Clarifies set boundaries—explicitly stating non-membership prevents ambiguity in formal arguments

Set-Builder Notation: {x | P(x)}

• Defines sets by properties rather than listing—$\{x \mid x > 0\}$ describes all positive numbers without enumeration
• The vertical bar means "such that"—read $\{x \mid P(x)\}$ as "the set of all $x$ such that $P(x)$ is true"
• Critical for infinite sets—you can't list all integers, but $\{n \mid n \in \mathbb{Z}\}$ captures them precisely

---

Subset Relationships

Understanding containment between sets is crucial for proofs and logical arguments. These symbols describe when one set "fits inside" another—a hierarchical relationship that forms the backbone of mathematical structure.

Subset: ⊆

• Every element of $A$ is also in $B$—written $A \subseteq B$, this allows for the possibility that $A = B$
• Every set is a subset of itself—$A \subseteq A$ is always true, a reflexive property you'll use in proofs
• The empty set is a subset of everything—$\emptyset \subseteq A$ for any set $A$, a fact that often appears in exam questions

Proper Subset: ⊂

• Subset but not equal—$A \subset B$ means all elements of $A$ are in $B$, but $B$ contains at least one element not in $A$
• Implies strict containment—if $A = \{1, 2\}$ and $B = \{1, 2, 3\}$, then $A \subset B$ but $A \not\subset A$
• Watch notation variations—some texts use $\subset$ for any subset and $\subsetneq$ for proper subsets; know your course's convention

---

Set Operations

These operations let you build new sets from existing ones. Mastering union, intersection, difference, and complement gives you the tools to manipulate sets algebraically—essential for solving problems and constructing proofs.

Union: ∪

• Combines all elements from both sets—$A \cup B$ contains everything in $A$, everything in $B$, with duplicates removed automatically
• Example: If $A = \{1, 2\}$ and $B = \{2, 3\}$, then $A \cup B = \{1, 2, 3\}$
• Corresponds to logical OR—$x \in A \cup B$ means $x \in A$ or $x \in B$ (or both)

Intersection: ∩

• Captures only shared elements—$A \cap B$ contains elements that appear in both $A$ and $B$
• Example: If $A = \{1, 2\}$ and $B = \{2, 3\}$, then $A \cap B = \{2\}$
• Corresponds to logical AND—$x \in A \cap B$ means $x \in A$ and $x \in B$

Set Difference: \

• Elements in one set but not the other—$A \setminus B$ (or $A - B$) contains elements of $A$ that aren't in $B$
• Example: If $A = \{1, 2, 3\}$ and $B = \{2, 3\}$, then $A \setminus B = \{1\}$
• Not commutative—$A \setminus B \neq B \setminus A$ in general, unlike union and intersection

Complement: $A^c$ or $A'$

• Everything not in the set—$A^c$ contains all elements of the universal set $U$ that aren't in $A$
• Example: If $U = \{1, 2, 3, 4\}$ and $A = \{1, 2\}$, then $A^c = \{3, 4\}$
• Requires a universal set—complement is meaningless without specifying what "everything" means in context

---

Special Sets

These sets have unique properties that make them foundational to set theory. Recognizing them and understanding their roles will help you avoid common errors and handle edge cases in proofs.

Empty Set: ∅ or { }

• Contains no elements—the unique set with cardinality zero, written $\emptyset$ or $\{\}$
• Subset of every set—$\emptyset \subseteq A$ is vacuously true for any $A$, a crucial fact for proofs
• Identity element for union—$A \cup \emptyset = A$; also $A \cap \emptyset = \emptyset$ for any set $A$

Universal Set: U

• Contains all elements under consideration—defines the "universe of discourse" for a given problem
• Context-dependent—in number theory $U$ might be $\mathbb{Z}$; in probability, it's the sample space
• Makes complements meaningful—$A^c = U \setminus A$, so always identify $U$ before computing complements

---

Advanced Constructions

These notations let you build more complex mathematical objects from sets—ordered pairs, all possible subsets, and size comparisons. They're essential for understanding relations, functions, and combinatorics.

Cartesian Product: ×

• Creates ordered pairs from two sets—$A \times B = \{(a, b) \mid a \in A, b \in B\}$
• Example: If $A = \{1, 2\}$ and $B = \{x, y\}$, then $A \times B = \{(1, x), (1, y), (2, x), (2, y)\}$
• Order matters here—$(1, x) \neq (x, 1)$, and $A \times B \neq B \times A$ unless $A = B$

Power Set: P(A)

• Set of all subsets—$\mathcal{P}(A)$ includes $\emptyset$, $A$ itself, and every subset in between
• Cardinality formula: If $|A| = n$, then $|\mathcal{P}(A)| = 2^n$—memorize this for quick calculations
• Example: If $A = \{1, 2\}$, then $\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$

Cardinality: |A|

• Counts elements in a set—$|A|$ gives the "size" of set $A$
• Example: If $A = \{1, 2, 3\}$, then $|A| = 3$; also $|\emptyset| = 0$
• Useful formulas: $|A \cup B| = |A| + |B| - |A \cap B|$ (inclusion-exclusion principle)

---

Quick Reference Table

---

Self-Check Questions

1. If $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$, compute $A \cup B$, $A \cap B$, and $A \setminus B$. Which operation corresponds to logical AND?

2. Explain the difference between $A \subseteq B$ and $A \subset B$. Give an example where $A \subseteq B$ is true but $A \subset B$ is false.

3. If $|A| = 3$, how many elements are in $\mathcal{P}(A)$? List all subsets of $A = \{a, b, c\}$ to verify your answer.

4. Compare and contrast set difference ($A \setminus B$) and complement ($A^c$). Under what conditions would $A \setminus B = A^c$?

5. Write the set of all even positive integers less than 10 using both roster notation and set-builder notation. Which notation would you use for "all even positive integers"? Why?