DISCLAIMER:–

THE CONTENT PROVIDED IN THIS ARTICLE IS FOR EDUCATIONAL AND INFORMATIONAL PURPOSES ONLY. WHILE EVERY EFFORT HAS BEEN MADE TO ENSURE THE ACCURACY AND CLARITY OF THE DEFINITIONS, THEOREMS, AND EXAMPLES RELATED TO SET THEORY. THE INFORMATION MAY NOT BE EXHAUSTIVE OR SUITED TO EVERY ACADEMIC CURRICULUM. READERS ARE ENCOURAGED TO CONSULT OFFICIAL TEXTBOOKS, ACADEMIC INSTRUCTORS OR SUBJECT EXPERTS FOR DETAILED GUIDANCE. THIS ARTICLE DOES NOT CONSTITUTE PROFESSIONAL OR ACADEMIC ADVICE AND SHOULD NOT BE USED AS A SOLE RESOURCE FOR EXAMINATION OR FORMAL STUDY PURPOSES. THE AUTHOR AND WEBSITE ARE NOT RESPONSIBLE FOR ANY ERRORS, OMISSIONS OR OUTCOMES RESULTING FROM THE USE OF THIS INFORMATION.

SET THEORY IS A FUNDAMENTAL PART OF MATHEMATICS AND FORMS THE BASIS FOR TOPICS SUCH AS LOGIC, PROBABILITY, ALGEBRA AND COMPUTER SCIENCE. THIS GUIDE WALKS YOU THROUGH SET THEORY FROM THE GROUND UP — STARTING WITH CORE CONCEPTS AND PROGRESSING TO THEOREMS AND ADVANCED OPERATIONS.

Table of Contents

1. WHAT IS A SET?

A SET IS A WELL-DEFINED COLLECTION OF DISTINCT OBJECTS, CONSIDERED AS AN OBJECT IN ITS OWN RIGHT.

NOTATION:-

A SET IS USUALLY DENOTED BY CAPITAL LETTERS: A,B,CA, B, CA,B,C

ELEMENTS ARE LISTED WITHIN CURLY BRACKETS: A={1,2,3}A = \{1, 2, 3\}A={1,2,3}

X∈AX \IN AX∈A: X IS AN ELEMENT OF A

Y∉AY \NOTIN AY∈/A: Y IS NOT AN ELEMENT OF A

TYPES OF SETS

TYPE	EXAMPLE
EMPTY SET (∅\EMPTYSET∅)	∅={}\EMPTYSET = \{\}∅={}
FINITE SET	A={2,4,6}A = \{2, 4, 6\}A={2,4,6}
INFINITE SET	B={X:X∈N}B = \{X : X \IN \MATHBB{N} \}B={X:X∈N}
EQUAL SETS	{1,2,3}={3,2,1}\{1,2,3\} = \{3,2,1\}{1,2,3}={3,2,1}
SUBSET	A⊆BA \SUBSETEQ BA⊆B
PROPER SUBSET	A⊂BA \SUBSET BA⊂B
UNIVERSAL SET (U)	INCLUDES ALL ELEMENTS UNDER CONSIDERATION
POWER SET (P(A)\MATHCAL{P}(A)P(A))	ALL SUBSETS OF A

SET OPERATIONS

1. UNION:-

A∪B={X:X∈A OR X∈B}A \CUP B = \{X : X \IN A \TEXT{ OR } X \IN B \}A∪B={X:X∈A OR X∈B}
EXAMPLE: {1,2}∪{2,3}={1,2,3}\{1,2\} \CUP \{2,3\} = \{1,2,3\}{1,2}∪{2,3}={1,2,3}

2. INTERSECTION:-

A∩B={X:X∈A AND X∈B}A \CAP B = \{X : X \IN A \TEXT{ AND } X \IN B \}A∩B={X:X∈A AND X∈B}
EXAMPLE: {1,2,3}∩{2,3,4}={2,3}\{1,2,3\} \CAP \{2,3,4\} = \{2,3\}{1,2,3}∩{2,3,4}={2,3}

3. SET DIFFERENCE:

A∖B={X:X∈A AND X∉B}A \SETMINUS B = \{X : X \IN A \TEXT{ AND } X \NOTIN B \}A∖B={X:X∈A AND X∈/B}
EXAMPLE: {1,2,3}∖{2}={1,3}\{1,2,3\} \SETMINUS \{2\} = \{1,3\}{1,2,3}∖{2}={1,3}

4. COMPLEMENT:

A′=U∖AA’ = U \SETMINUS AA′=U∖A
EXAMPLE: IF U={1,2,3,4},A={1,2}U = \{1,2,3,4\}, A = \{1,2\}U={1,2,3,4},A={1,2}, THEN A′={3,4}A’ = \{3,4\}A′={3,4}

5. SYMMETRIC DIFFERENCE:-

AΔB=(A∖B)∪(B∖A)A \DELTA B = (A \SETMINUS B) \CUP (B \SETMINUS A)AΔB=(A∖B)∪(B∖A)

4. VENN DIAGRAMS

A VISUAL REPRESENTATION OF SETS:

CIRCLES REPRESENT SETS

OVERLAPPING AREAS SHOW INTERSECTIONS

USEFUL FOR SOLVING WORD PROBLEMS INVOLVING UNIONS AND INTERSECTIONS

5. IMPORTANT SET THEORY LAWS

IDENTITY LAWS:

A∪∅=AA \CUP \EMPTYSET = AA∪∅=A

A∩U=AA \CAP U = AA∩U=A

DOMINATION LAWS:-

A∪U=UA \CUP U = UA∪U=U

A∩∅=∅A \CAP \EMPTYSET = \EMPTYSETA∩∅=∅

IDEMPOTENT LAWS:-

A∪A=AA \CUP A = AA∪A=A

A∩A=AA \CAP A = AA∩A=A

COMPLEMENT LAWS:-

A∪A′=UA \CUP A’ = UA∪A′=U

A∩A′=∅A \CAP A’ = \EMPTYSETA∩A′=∅

DE MORGAN’S LAWS:-

(A∪B)′=A′∩B′(A \CUP B)’ = A’ \CAP B'(A∪B)′=A′∩B′

(A∩B)′=A′∪B′(A \CAP B)’ = A’ \CUP B'(A∩B)′=A′∪B′

DISTRIBUTIVE LAWS:-

A∪(B∩C)=(A∪B)∩(A∪C)A \CUP (B \CAP C) = (A \CUP B) \CAP (A \CUP C)A∪(B∩C)=(A∪B)∩(A∪C)

A∩(B∪C)=(A∩B)∪(A∩C)A \CAP (B \CUP C) = (A \CAP B) \CUP (A \CAP C)A∩(B∪C)=(A∩B)∪(A∩C)

CARDINALITY (SIZE OF SET)

∣A∣|A|∣A∣: NUMBER OF ELEMENTS IN SET A

IF A={1,2,3}A = \{1,2,3\}A={1,2,3}, THEN ∣A∣=3|A| = 3∣A∣=3

POWER SET FORMULA:

∣P(A)∣=2∣A∣|\MATHCAL{P}(A)| = 2^{|A|}∣P(A)∣=2∣A∣

WORD PROBLEM EXAMPLE

Q: IN A CLASS OF 50 STUDENTS:

30 LIKE MATH

25 LIKE SCIENCE

10 LIKE BOTH
HOW MANY LIKE ONLY MATH?

SOLUTION:
ONLY MATH = 30−10=2030 – 10 = 2030−10=20

ADVANCED SET CONCEPTS:-

SET BUILDER NOTATION:-

A={X:X IS A PRIME NUMBER}A = \{X : X \TEXT{ IS A PRIME NUMBER} \}A={X:X IS A PRIME NUMBER}

INDEXED SETS:-

AIA_IAI: COLLECTION OF SETS INDEXED BY I

⋃I=1NAI\BIGCUP_{I=1}^N A_I⋃I=1NAI, ⋂I=1NAI\BIGCAP_{I=1}^N A_I⋂I=1NAI

INFINITE SETS:

COUNTABLY INFINITE: N,Z\MATHBB{N}, \MATHBB{Z}N,Z

UNCOUNTABLY INFINITE: R\MATHBB{R}R

CARTESIAN PRODUCT:-

A×B={(A,B):A∈A,B∈B}A \TIMES B = \{(A,B) : A \IN A, B \IN B\}A×B={(A,B):A∈A,B∈B}

EXAMPLE:–
IF A={1,2},B={X,Y}A = \{1,2\}, B = \{X,Y\}A={1,2},B={X,Y}, THEN
A×B={(1,X),(1,Y),(2,X),(2,Y)}A \TIMES B = \{(1,X), (1,Y), (2,X), (2,Y)\}A×B={(1,X),(1,Y),(2,X),(2,Y)}

APPLICATIONS OF SET THEORY

DATABASE QUERIES (SQL, LOGIC OPERATORS)

COMPUTER SCIENCE (AUTOMATA, LANGUAGES)

PROBABILITY & STATISTICS

LOGIC AND PROOFS

VENN DIAGRAM-BASED ANALYTICS

PRACTICE QUESTIONS

LIST THE POWER SET OF {A,B}\{A, B\}{A,B}

IF A={1,2,3},B={3,4,5}A = \{1,2,3\}, B = \{3,4,5\}A={1,2,3},B={3,4,5}, FIND A∪B,A∩BA \CUP B, A \CAP BA∪B,A∩B

PROVE: A∪(A∩B)=AA \CUP (A \CAP B) = AA∪(A∩B)=A

CONCLUSION

SET THEORY – ADVANCED CONCEPTS, THEOREMS & APPLICATIONS (PART 2)

CARTESIAN PRODUCTS & RELATIONS

DEFINITION:

THE CARTESIAN PRODUCT OF TWO SETS A AND B, DENOTED A×BA \TIMES BA×B, IS THE SET OF ALL ORDERED PAIRS (A,B)(A, B)(A,B), WHERE A∈AA \IN AA∈A AND B∈BB \IN BB∈B.

FORMULA:-

IF ∣A∣=M|A| = M∣A∣=M AND ∣B∣=N|B| = N∣B∣=N, THEN

∣A×B∣=M×N|A \TIMES B| = M \TIMES N∣A×B∣=M×N

EXAMPLE:-

LET A={1,2},B={X,Y}A = \{1,2\}, B = \{X,Y\}A={1,2},B={X,Y}
THEN A×B={(1,X),(1,Y),(2,X),(2,Y)}A \TIMES B = \{(1,X), (1,Y), (2,X), (2,Y)\}A×B={(1,X),(1,Y),(2,X),(2,Y)}

ADVANCED THEOREMS IN SET THEORY

THEOREM 1:-

A∪(B∩C)=(A∪B)∩(A∪C)A \CUP (B \CAP C) = (A \CUP B) \CAP (A \CUP C)A∪(B∩C)=(A∪B)∩(A∪C)

PROOF SKETCH:-

USE ELEMENT-WISE ARGUMENT. SHOW BOTH LHS ⊆ RHS AND RHS ⊆ LHS.

THEOREM 2:-

IF A⊆BA \SUBSETEQ BA⊆B AND B⊆CB \SUBSETEQ CB⊆C, THEN A⊆CA \SUBSETEQ CA⊆C

PROOF:–

TRANSITIVE PROPERTY OF SETS.

THEOREM 3:-

AΔB=(A∪B)∖(A∩B)A \DELTA B = (A \CUP B) \SETMINUS (A \CAP B)AΔB=(A∪B)∖(A∩B)

EXPLANATION:–

SYMMETRIC DIFFERENCE REMOVES COMMON ELEMENTS.

THEOREM 4:-

(A∖B)∪(A∩B)=A(A \SETMINUS B) \CUP (A \CAP B) = A(A∖B)∪(A∩B)=A

PROOF IDEA:-

EVERY ELEMENT IN A IS EITHER IN B OR NOT IN B.

NDEXED FAMILY OF SETS

DEFINITION:-

AN INDEXED FAMILY OF SETS IS A COLLECTION {AI}I∈I\{A_I\}_{I \IN I}{AI}I∈I, WHERE EACH AIA_IAI IS A SET LABELED BY AN INDEX III.

OPERATIONS:

⋃I∈IAI\BIGCUP_{I \IN I} A_I⋃I∈IAI: UNION OVER ALL INDEXED SETS

⋂I∈IAI\BIGCAP_{I \IN I} A_I⋂I∈IAI: INTERSECTION OVER ALL INDEXED SETS

EXAMPLE:

LET AN={N,N+1,N+2}A_N = \{N, N+1, N+2\}AN={N,N+1,N+2}
THEN:

A1={1,2,3}A_1 = \{1,2,3\}A1={1,2,3}

A2={2,3,4}A_2 = \{2,3,4\}A2={2,3,4}

A3={3,4,5}A_3 = \{3,4,5\}A3={3,4,5}
AND: ⋂N=13AN={3}\BIGCAP_{N=1}^{3} A_N = \{3\}⋂N=13AN={3}

COUNTABILITY AND INFINITY

COUNTABLE SETS:-

A SET IS COUNTABLE IF ITS ELEMENTS CAN BE PUT IN ONE-TO-ONE CORRESPONDENCE WITH THE NATURAL NUMBERS N\MATHBB{N}N.

EXAMPLES: N,Z,Q\MATHBB{N}, \MATHBB{Z}, \MATHBB{Q}N,Z,Q

UNCOUNTABLE SETS:-

SETS THAT CANNOT BE LISTED COMPLETELY, LIKE REAL NUMBERS R\MATHBB{R}R, ARE UNCOUNTABLE.

CANTOR’S DIAGONALIZATION PROOF SHOWS R\MATHBB{R}R IS UNCOUNTABLE.

APPLICATIONS OF SET THEORY IN VARIOUS FIELDS

FIELD	APPLICATION EXAMPLE
COMPUTER SCIENCE	DATABASE QUERIES (SQL), ALGORITHMS, FORMAL LANGUAGES
LOGIC	SYMBOLIC LOGIC, PREDICATE LOGIC
PROBABILITY	SAMPLE SPACES, EVENTS, INTERSECTIONS, AND UNIONS
ARTIFICIAL INTELLIGENCE	KNOWLEDGE REPRESENTATION, FUZZY SETS
DISCRETE MATH	GRAPH THEORY, RELATIONS, AUTOMATA THEORY

REAL-WORLD WORD PROBLEM (ADVANCED)

Q: IN A GROUP OF 200 STUDENTS:

120 LIKE MATH

80 LIKE PHYSICS

60 LIKE BOTH

20 LIKE NEITHER

HOW MANY LIKE ONLY MATH OR ONLY PHYSICS?

✅ SOLUTION:
LET:

ONLY MATH = 120−60=60120 – 60 = 60120−60=60
ONLY PHYSICS = 80−60=2080 – 60 = 2080−60=20
BOTH = 60
NEITHER = 20

TOTAL = 60 + 20 + 60 + 20 = 160 + 20 = 180?
BUT GROUP HAS 200 STUDENTS. SO ONLY MATH OR PHYSICS = 200−60−20=120200 – 60 – 20 = 120200−60−20=120

✅ ANSWER: 80 STUDENTS LIKE ONLY ONE SUBJECT.

PROOF-BASED CHALLENGE

PROVE: A∪(A∩B)=AA \CUP (A \CAP B) = AA∪(A∩B)=A

✅ PROOF:
LET X∈A∪(A∩B)X \IN A \CUP (A \CAP B)X∈A∪(A∩B)

THEN EITHER X∈AX \IN AX∈A, OR X∈A∩B⇒X∈AX \IN A \CAP B \RIGHTARROW X \IN AX∈A∩B⇒X∈A

HENCE X∈AX \IN AX∈A, SO A∪(A∩B)⊆AA \CUP (A \CAP B) \SUBSETEQ AA∪(A∩B)⊆A

ALSO, A⊆A∪(A∩B)A \SUBSETEQ A \CUP (A \CAP B)A⊆A∪(A∩B), SO THEY ARE EQUAL.

✅ HENCE, PROVED.

PRACTICE SET (ADVANCED)

PROVE OR DISPROVE: AΔB=(A∪B)∖(A∩B)A \DELTA B = (A \CUP B) \SETMINUS (A \CAP B)AΔB=(A∪B)∖(A∩B)
LET A={X:X∈N,X IS EVEN},B={X∈N:X DIVISIBLE BY 3}A = \{X : X \IN \MATHBB{N}, X \TEXT{ IS EVEN} \}, B = \{X \IN \MATHBB{N} : X \TEXT{ DIVISIBLE BY 3} \}A={X:X∈N,X IS EVEN},B={X∈N:X DIVISIBLE BY 3}. FIND A∩BA \CAP BA∩B
DRAW VENN DIAGRAMS FOR A∪(B∩C)A \CUP (B \CAP C)A∪(B∩C) AND (A∪B)∩(A∪C)(A \CUP B) \CAP (A \CUP C)(A∪B)∩(A∪C)
PROVE DE MORGAN’S LAWS USING VENN DIAGRAMS
LIST THE POWER SET OF {A,B,C}\{A, B, C\}{A,B,C}

SET THEORY – EXPERT CONCEPTS, PROOFS & SPECIAL TOPICS (PART 3)

SET BUILDER NOTATION & INTERVAL NOTATION

SET BUILDER NOTATION:

USED TO DEFINE SETS USING A RULE OR PROPERTY.

EXAMPLES:

A={X∈R:X>0}A = \{X \IN \MATHBB{R} : X > 0\}A={X∈R:X>0}

B={X:X IS AN EVEN NUMBER AND X<20}B = \{X : X \TEXT{ IS AN EVEN NUMBER AND } X < 20\}B={X:X IS AN EVEN NUMBER AND X<20}

INTERVAL NOTATION:

USED TO REPRESENT SUBSETS OF REAL NUMBERS.

INTERVAL TYPE	NOTATION	MEANING
OPEN	(A,B)(A, B)(A,B)	A<X<BA < X < BA<X<B
CLOSED	[A,B][A, B][A,B]	A≤X≤BA \LE X \LE BA≤X≤B
HALF-OPEN (LEFT)	[A,B)[A, B)[A,B)	A≤X<BA \LE X < BA≤X<B
HALF-OPEN (RIGHT)	(A,B](A, B](A,B]	A<X≤BA < X \LE BA<X≤B
INFINITE	(−∞,A)(-\INFTY, A)(−∞,A)	X<AX < AX<A

LOGICAL CONNECTIVES & SETS

SET OPERATIONS CAN BE MAPPED TO LOGICAL OPERATIONS:

SET OPERATION	LOGIC EQUIVALENT
A∪BA \CUP BA∪B	P∨QP \LOR QP∨Q
A∩BA \CAP BA∩B	P∧QP \LAND QP∧Q
A′A’A′	¬P\NEG P¬P
A⊆BA \SUBSETEQ BA⊆B	P⇒QP \RIGHTARROW QP⇒Q

FUZZY SETS:-

DEFINITION:

A FUZZY SET IS A SET WHERE ELEMENTS HAVE DEGREES OF MEMBERSHIP, REPRESENTED BY VALUES IN [0,1][0, 1][0,1].

EXAMPLE:-

TRADITIONAL SET: A={COLD,WARM}A = \{COLD, WARM\}A={COLD,WARM}

FUZZY SET: A={COLD/0.9,COOL/0.6,WARM/0.3}A = \{\TEXT{COLD}/0.9, \TEXT{COOL}/0.6, \TEXT{WARM}/0.3\}A={COLD/0.9,COOL/0.6,WARM/0.3}

USED IN:-

ARTIFICIAL INTELLIGENCE

CONTROL SYSTEMS

DECISION MAKING

MLTISETS (BAGS)

DEFINITION:-

A MULTISET ALLOWS MULTIPLE INSTANCES OF ELEMENTS.

EXAMPLE:–
A={A,A,B,B,B}A = \{A, A, B, B, B\}A={A,A,B,B,B}

NOTATIONS LIKE A(2),B(3)A^{(2)}, B^{(3)}A(2),B(3) CAN BE USED.

USED IN:

DATABASES

COUNTING ALGORITHMS

CHEMISTRY (MOLECULES)

ADVANCED SET-THEORETIC IDENTITIES (PROOF STYLE)

THEOREM 5: A∖(B∪C)=(A∖B)∩(A∖C)A \SETMINUS (B \CUP C) = (A \SETMINUS B) \CAP (A \SETMINUS C)A∖(B∪C)=(A∖B)∩(A∖C)

PROOF:-

LET X∈A∖(B∪C)⇒X∈AX \IN A \SETMINUS (B \CUP C) \RIGHTARROW X \IN AX∈A∖(B∪C)⇒X∈A AND X∉B∪CX \NOTIN B \CUP CX∈/B∪C
⇒X∉B\RIGHTARROW X \NOTIN B⇒X∈/B AND X∉C⇒X∈(A∖B)∩(A∖C)X \NOTIN C \RIGHTARROW X \IN (A \SETMINUS B) \CAP (A \SETMINUS C)X∈/C⇒X∈(A∖B)∩(A∖C)

HENCE BOTH SETS ARE EQUAL.

RUSSELL’S PARADOX (SET THEORY FOUNDATION)

PARADOX:-

LET R={X:X∉X}R = \{X : X \NOTIN X\}R={X:X∈/X}.
DOES R∈RR \IN RR∈R?

IF YES → R∉RR \NOTIN RR∈/R

IF NO → R∈RR \IN RR∈R

THIS LEADS TO A CONTRADICTION.
✅ THIS PARADOX SHOWS LIMITATIONS IN NAIVE SET THEORY.

LED TO DEVELOPMENT OF:

ZERMELO-FRAENKEL SET THEORY (ZF)

AXIOMATIC SET THEORY

ZERMELO-FRAENKEL AXIOMS (OVERVIEW)

KEY AXIOMS TO AVOID CONTRADICTIONS IN SET THEORY:

AXIOM OF EXTENSIONALITY
AXIOM OF PAIRING
AXIOM OF UNION
AXIOM OF POWER SET
AXIOM OF INFINITY
AXIOM OF REPLACEMENT
AXIOM OF CHOICE (ZFC)

EACH AXIOM RESTRICTS HOW SETS ARE CONSTRUCTED AND AVOIDS PARADOXES.

PRACTICAL APPLICATIONS IN COMPUTER SCIENCE

CONCEPT	APPLICATION EXAMPLE
POWER SETS	GENERATING COMBINATIONS
SUBSET CHECKING	PERMISSIONS IN SECURITY & ROLE MANAGEMENT
SET DIFFERENCE	DATA COMPARISON & CHANGE DETECTION
FUZZY SETS	AI, ROBOTICS, RECOMMENDATION ENGINES
BOOLEAN ALGEBRA	DIGITAL CIRCUIT DESIGN, LOGIC GATES

CHALLENGE PROBLEMS (EXPERT LEVEL)

PROVE: A∩(B∪C)=(A∩B)∪(A∩C)A \CAP (B \CUP C) = (A \CAP B) \CUP (A \CAP C)A∩(B∪C)=(A∩B)∪(A∩C)
GIVEN: A={X∈Z:X≡1MOD 3},B={X∈Z:X≡2MOD 3}A = \{X \IN \MATHBB{Z} : X \EQUIV 1 \MOD 3\}, B = \{X \IN \MATHBB{Z} : X \EQUIV 2 \MOD 3\}A={X∈Z:X≡1MOD3},B={X∈Z:X≡2MOD3}, FIND A∩BA \CAP BA∩B
IF A⊂BA \SUBSET BA⊂B AND ∣A∣=∣B∣|A| = |B|∣A∣=∣B∣, WHAT CAN YOU CONCLUDE?
EXPLAIN WHY THE POWER SET OF A COUNTABLY INFINITE SET IS UNCOUNTABLE.
APPLY DE MORGAN’S LAW TO THE EXPRESSION: (A∪B∪C)′(A \CUP B \CUP C)'(A∪B∪C)′

FINAL THOUGHTS:-

SET THEORY EVOLVES FROM INTUITIVE COLLECTIONS OF OBJECTS TO RIGOROUS LOGICAL SYSTEMS THAT UNDERPIN MUCH OF MODERN MATHEMATICS AND COMPUTATION. MASTERY OF SET THEORY NOT ONLY ENHANCES ANALYTICAL SKILLS BUT ALSO ENABLES DEEPER UNDERSTANDING IN SUBJECTS LIKE LOGIC, ALGORITHMS, DATABASES AND BEYOND.

SET THEORY – EXPERT TOPICS, STRUCTURES, AND LOGICAL FOUNDATIONS (PART 4)

POWER SET AND HIGHER CARDINALITIES

POWER SET DEFINITION:-

FOR ANY SET AAA, ITS POWER SET P(A)\MATHCAL{P}(A)P(A) IS THE SET OF ALL SUBSETS OF AAA.

IF ∣A∣=N|A| = N∣A∣=N, THEN ∣P(A)∣=2N|\MATHCAL{P}(A)| = 2^N∣P(A)∣=2N

CANTOR’S THEOREM:-

FOR ANY SET AAA,

∣P(A)∣>∣A∣|\MATHCAL{P}(A)| > |A|∣P(A)∣>∣A∣

THIS SHOWS THAT THERE IS NO SURJECTION FROM A SET TO ITS POWER SET — LEADING TO INFINITE HIERARCHY OF INFINITIES.

ORDINAL NUMBERS (TRANSFINITE ORDERING)

DEFINITION:-

ORDINALS EXTEND NATURAL NUMBERS TO DESCRIBE ORDER TYPES OF WELL-ORDERED SETS.

EXAMPLES:

0,1,2,…,Ω,Ω+1,Ω⋅2,ΩΩ0, 1, 2, \DOTS, \OMEGA, \OMEGA + 1, \OMEGA \CDOT 2, \OMEGA^\OMEGA0,1,2,…,Ω,Ω+1,Ω⋅2,ΩΩ

Ω\OMEGAΩ: THE FIRST LIMIT ORDINAL, REPRESENTING THE SMALLEST INFINITE ORDER.

ORDINALS INCLUDE SUCCESSORS AND LIMITS.

CARDINAL NUMBERS (MEASURING SIZE OF SETS)

SYMBOL	DESCRIPTION	EXAMPLE
ℵ0\ALEPH_0ℵ0	COUNTABLE INFINITY	N,Z\MATHBB{N}, \MATHBB{Z}N,Z
ℵ1\ALEPH_1ℵ1	NEXT LARGER INFINITY	HYPOTHETICALLY R\MATHBB{R}R UNDER CH
CCC	CARDINALITY OF R\MATHBB{R}R	CONTINUUM

SCHRÖDER–BERNSTEIN THEOREM:-

IF A⊆BA \SUBSETEQ BA⊆B AND B⊆AB \SUBSETEQ AB⊆A, THEN ∣A∣=∣B∣|A| = |B|∣A∣=∣B∣

THIS HELPS PROVE SET EQUIVALENCES WITHOUT BUILDING BIJECTIONS EXPLICITLY.

ALGEBRA OF SETS (BOOLEAN ALGEBRA)

SET THEORY CAN BE FORMALIZED AS A BOOLEAN ALGEBRA.

OPERATION	SET THEORY	LOGIC EQUIVALENT
UNION	A∪BA \CUP BA∪B	OR
INTERSECTION	A∩BA \CAP BA∩B	AND
COMPLEMENT	ACA^CAC OR A′A’A′	NOT
DIFFERENCE	A−BA – BA−B	AND NOT
SYM. DIFF	AΔBA \DELTA BAΔB	XOR

LAWS:

IDEMPOTENT: A∪A=AA \CUP A = AA∪A=A, A∩A=AA \CAP A = AA∩A=A

DOMINATION: A∪U=UA \CUP U = UA∪U=U, A∩∅=∅A \CAP \EMPTYSET = \EMPTYSETA∩∅=∅

INVOLUTION: (AC)C=A(A^C)^C = A(AC)C=A

DOUBLE COMPLEMENT, ABSORPTION, DISTRIBUTIVE, DE MORGAN’S LAWS…

FIXED POINT THEOREM (KNASTER–TARSKI)

IN ORDER THEORY AND SET THEORY:

ANY MONOTONIC FUNCTION FFF ON A COMPLETE LATTICE HAS AT LEAST ONE FIXED POINT.
THAT IS, F(X)=XF(X) = XF(X)=X

USED IN:

PROGRAMMING LANGUAGE SEMANTICS

RECURSION THEORY

DOMAIN THEORY

FILTERS AND ULTRAFILTERS (ADVANCED LOGIC)

FILTER:

A FILTER ON A SET XXX IS A COLLECTION F⊆P(X)\MATHCAL{F} \SUBSETEQ \MATHCAL{P}(X)F⊆P(X) SUCH THAT:

∅∉F\EMPTYSET \NOTIN \MATHCAL{F}∅∈/F

A,B∈F⇒A∩B∈FA, B \IN \MATHCAL{F} \RIGHTARROW A \CAP B \IN \MATHCAL{F}A,B∈F⇒A∩B∈F

A∈F,A⊆B⇒B∈FA \IN \MATHCAL{F}, A \SUBSETEQ B \RIGHTARROW B \IN \MATHCAL{F}A∈F,A⊆B⇒B∈F

ULTRAFILTER:

A MAXIMAL FILTER WHERE FOR ANY A⊆XA \SUBSETEQ XA⊆X, EITHER A∈FA \IN \MATHCAL{F}A∈F OR AC∈FA^C \IN \MATHCAL{F}AC∈F

USED IN:

MODEL THEORY

NONSTANDARD ANALYSIS

COMPACTNESS THEOREMS

AXIOM OF CHOICE (AC)

STATEMENT:

GIVEN ANY COLLECTION OF NON-EMPTY SETS, IT IS POSSIBLE TO CHOOSE ONE ELEMENT FROM EACH SET, EVEN IF THE COLLECTION IS INFINITE.

IMPLICATIONS:-

EQUIVALENT TO: ZORN’S LEMMA, WELL-ORDERING THEOREM

REQUIRED IN PROOFS INVOLVING EXISTENCE WITHOUT CONSTRUCTION

WELL-ORDERING PRINCIPLE

EVERY SET CAN BE WELL-ORDERED IF YOU ACCEPT THE AXIOM OF CHOICE.

WELL-ORDERING: EVERY NON-EMPTY SUBSET HAS A LEAST ELEMENT.

APPLICATION: PROVING PROPERTIES BY TRANSFINITE INDUCTION.

TOPOLOGY AND SET THEORY

SET THEORY FORMS THE FOUNDATION OF TOPOLOGICAL SPACES:

OPEN SETS, UNIONS, INTERSECTIONS

BASIS FOR TOPOLOGY: A COLLECTION OF OPEN SETS SATISFYING CERTAIN CONDITIONS

POWER SETS FORM TOPOLOGIES UNDER CONSTRAINTS (E.G., DISCRETE, COFINITE)

FOUNDATIONS OF MATHEMATICS

SET THEORY UNDERLIES PEANO ARITHMETIC, REAL ANALYSIS, NUMBER THEORY THROUGH:

CONSTRUCTION OF N,Z,Q,R,C\MATHBB{N}, \MATHBB{Z}, \MATHBB{Q}, \MATHBB{R}, \MATHBB{C}N,Z,Q,R,C

FUNCTIONS AS SETS OF ORDERED PAIRS

RELATIONS, ORDERINGS, BIJECTIONS

ADVANCED CHALLENGE PROBLEMS

PROVE OR DISPROVE: Ω+1=1+Ω\OMEGA + 1 = 1 + \OMEGAΩ+1=1+Ω
SHOW THAT THE SET OF ALL SUBSETS OF N\MATHBB{N}N IS UNCOUNTABLE
DEFINE A FILTER AND ULTRAFILTER ON A FINITE SET
PROVE THE SCHRÖDER–BERNSTEIN THEOREM
SHOW THAT NOT ALL FUNCTIONS FROM N→N\MATHBB{N} \RIGHTARROW \MATHBB{N}N→N ARE COMPUTABLE

SUMMARY (PART 4)

YOU NOW HAVE ACCESS TO THE DEEP STRUCTURAL AND LOGICAL ARCHITECTURE OF SET THEORY, FROM THE INTUITIVE TO THE INFINITE, FROM OPERATIONS TO PARADOXES, FROM FUNCTIONS TO FOUNDATIONS. THIS FORMS THE BASIS OF NOT JUST MATHEMATICAL THEORY BUT OF MODERN LOGIC, COMPUTATION, AND PROOF SYSTEMS.

SET THEORY – RESEARCH-LEVEL TOPICS AND PHILOSOPHICAL FOUNDATIONS (PART 5)

DESCRIPTIVE SET THEORY

FOCUS:

STUDY OF SETS OF REAL NUMBERS AND OTHER POLISH SPACES (COMPLETE SEPARABLE METRIC SPACES), PARTICULARLY IN TOPOLOGY AND ANALYSIS.

KEY TERMS:

BOREL SETS:–

GENERATED FROM OPEN SETS USING COUNTABLE UNIONS/INTERSECTIONS.

ANALYTIC SETS:–

CONTINUOUS IMAGE OF BOREL SETS.

PROJECTIVE HIERARCHY:–

CLASSIFICATION OF SETS BEYOND BOREL.

APPLICATIONS:

REAL ANALYSIS, MEASURE THEORY, FUNCTIONAL ANALYSIS, AND LOGIC.

CATEGORY THEORY AND SETS

CATEGORIES:

IN CATEGORY THEORY, SETS ARE PART OF THE CATEGORY SET, WHERE:

OBJECTS = SETS

MORPHISMS = FUNCTIONS BETWEEN SETS

CATEGORY SET IS A CONCRETE CATEGORY WITH:-

IDENTITY MORPHISMS

COMPOSITION OF MORPHISMS

FUNCTORS AND NATURAL TRANSFORMATIONS CONNECT SET THEORY TO ALGEBRA, TOPOLOGY, AND PROGRAMMING LANGUAGES (E.G., HASKELL).

FORCING (ADVANCED SET THEORY)

DEFINITION:

FORCING IS A METHOD DEVELOPED BY PAUL COHEN TO SHOW THAT CERTAIN STATEMENTS (LIKE THE CONTINUUM HYPOTHESIS) ARE INDEPENDENT OF ZFC.

CORE IDEA:

EXTEND A MODEL OF ZFC TO A BIGGER MODEL USING A PARTIALLY ORDERED SET AND GENERIC FILTERS.

RESULT:-

CAN CREATE MODELS WHERE CH IS TRUE OR FALSE.

USED IN:

PROVING INDEPENDENCE

EXPLORING ALTERNATE MATHEMATICAL UNIVERSES

MODELS OF SET THEORY

WHAT IS A MODEL?

A MODEL OF SET THEORY IS A COLLECTION OF SETS THAT SATISFIES THE AXIOMS OF ZFC.

TYPES:

STANDARD MODELS:-

INTENDED INTERPRETATION OF ZFC

NON-STANDARD MODELS:-

MAY CONTAIN “EXTRA” ELEMENTS OR STRUCTURES

INNER MODELS:-

SUBSTRUCTURES LIKE GÖDEL’S CONSTRUCTIBLE UNIVERSE LLL

CONSTRUCTIBLE UNIVERSE (GÖDEL’S LLL)

DEFINITION:

THE CLASS OF SETS CONSTRUCTIBLE FROM EARLIER SETS USING DEFINABLE OPERATIONS.

PROPERTIES:-

LLL IS A MODEL OF ZFC + GCH

PROVES RELATIVE CONSISTENCY OF GCH AND AC WITH ZF

IMPLICATION:-

CH IS CONSISTENT WITH (BUT NOT PROVABLE FROM) ZFC.

LARGE CARDINALS

LARGE CARDINALS:

INFINITE CARDINALS WITH EXTRA PROPERTIES NOT PROVABLE IN ZFC, SUCH AS:

INACCESSIBLE CARDINALS

MEASURABLE CARDINALS

WOODIN CARDINALS

THESE EXTEND THE HIERARCHY OF INFINITIES AND HAVE IMPLICATIONS FOR CONSISTENCY AND DETERMINACY.

GÖDEL’S INCOMPLETENESS THEOREMS (IMPACT ON SET THEORY)

FIRST THEOREM:-

ANY CONSISTENT FORMAL SYSTEM THAT IS EXPRESSIVE ENOUGH TO INCLUDE ARITHMETIC CANNOT PROVE ALL TRUTHS ABOUT THE ARITHMETIC OF NATURAL NUMBERS.

SECOND THEOREM:-

SUCH A SYSTEM CANNOT PROVE ITS OWN CONSISTENCY.

APPLIED TO ZFC, THIS MEANS:
ZFC CANNOT PROVE ITS OWN CONSISTENCY OR COMPLETENESS.

DETERMINACY AND GAMES

IN DESCRIPTIVE SET THEORY, MANY RESULTS RELY ON GAME THEORY AND THE CONCEPT OF DETERMINACY.

A GAME IS DETERMINED IF ONE PLAYER HAS A WINNING STRATEGY.

THE AXIOM OF DETERMINACY (AD) CONTRADICTS THE AXIOM OF CHOICE, BUT GIVES STRONG RESULTS IN ANALYSIS.

SET-THEORETIC TOPOLOGY

THIS AREA BLENDS TOPOLOGY AND SET THEORY.

KEY CONCEPTS:

LINDELÖF SPACES

COMPACTNESS AND FILTERS

MOORE SPACES, PARACOMPACTNESS

USE OF CARDINAL FUNCTIONS (E.G., WEIGHT, CHARACTER)

APPLICATIONS IN THEORETICAL COMPUTER SCIENCE

SET THEORY UNDERPINS:

LAMBDA CALCULUS & TYPE THEORY

AUTOMATA & FORMAL LANGUAGES

DECISION PROBLEMS & COMPUTABILITY

DOMAIN THEORY (SEMANTICS OF PROGRAMMING LANGUAGES)

ALTERNATIVE SET THEORIES

SYSTEM	FOCUS
ZFC	STANDARD FOUNDATION
ZF WITHOUT AC	CONSTRUCTIVIST/INTUITIONIST APPROACHES
NF (QUINE)	ALLOWS UNIVERSAL SET, NO PARADOX
TST (TYPE THEORY)	USED IN LOGIC AND CS FOUNDATIONS
MK (MORSE–KELLEY)	EXTENDS ZFC WITH CLASS THEORY

ULTIMATE CHALLENGE PROBLEMS:-

CONSTRUCT A MODEL OF ZF WHERE THE AXIOM OF CHOICE FAILS.
SHOW THAT NO LARGEST CARDINAL CAN EXIST (HINT: CONSIDER POWER SET).
DEMONSTRATE HOW FORCING CAN CHANGE THE TRUTH OF CH.
COMPARE TWO MODELS OF ZF + AC WITH AND WITHOUT GCH.
EXPLAIN WHY “THE SET OF ALL SETS” LEADS TO CONTRADICTION (RUSSELL’S PARADOX) — CAN THIS BE AVOIDED IN NF OR MK?

SET THEORY – LOGICAL LIMITS, PHILOSOPHY, AND METAMATHEMATICS (PART 6)

INDEPENDENCE RESULTS IN SET THEORY

SET THEORY INVESTIGATES STATEMENTS INDEPENDENT OF ZFC:

CLASSIC INDEPENDENT STATEMENTS:

CONTINUUM HYPOTHESIS (CH)

AXIOM OF DETERMINACY (AD)

SUSLIN’S HYPOTHESIS

WHITEHEAD’S PROBLEM

MEASURABLE CARDINALS

THESE REQUIRE BUILDING MODELS OF ZFC + STATEMENT AND ZFC + NEGATION USING TOOLS LIKE FORCING AND INNER MODELS.

INNER MODELS AND CORE MODELS

INNER MODELS:

A SUBCLASS OF THE UNIVERSE OF SETS THAT SATISFIES ZFC AND CONTAINS ALL ORDINALS.

EXAMPLES: LLL, L[U]L[U]L[U], CORE MODELS LIKE KKK

INNER MODELS HELP PROVE CONSISTENCY OF LARGE CARDINALS WITHOUT ASSUMING THEM.

CORE MODEL THEORY:-

STUDIES CANONICAL INNER MODELS THAT CAN SIMULATE LARGE CARDINALS WITHOUT ASSUMING THEIR ACTUAL EXISTENCE.

SET THEORY WITHOUT INFINITY

WHAT IF WE OMIT THE AXIOM OF INFINITY FROM ZF?

RESULT: THE THEORY NOW DESCRIBES FINITE SETS ONLY.

KNOWN AS FINITE SET THEORY (FST)

USED IN BOUNDED ARITHMETIC, FINITE MODEL THEORY, AND CERTAIN AREAS OF CS.

REFLECTION PRINCIPLES

DEFINITION:-

A REFLECTION PRINCIPLE STATES THAT ANY PROPERTY OF THE UNIVERSE OF SETS ALSO HOLDS IN SOME SMALLER SET OR CLASS.

EXAMPLE:
FOR EVERY FORMULA Φ\PHIΦ, THERE IS A TRANSITIVE SET MMM SUCH THAT (M,∈)⊨Φ(M, \IN) \MODELS \PHI(M,∈)⊨Φ

USED TO JUSTIFY:

INACCESSIBLE CARDINALS

INDESCRIBABLE CARDINALS

LARGE CARDINAL AXIOMS – DEEP DIVE

EXAMPLES BY STRENGTH:

TYPE	PROPERTY
INACCESSIBLE	NOT REACHABLE BY POWER SET / SUCCESSOR
MAHLO	INACCESSIBLES BELOW IT FORM A STATIONARY SET
MEASURABLE	EXISTS A NON-TRIVIAL Κ\KAPPAΚ-ADDITIVE 0-1 MEASURE
STRONG	EMBEDDING FROM VVV TO MMM WHERE CRITICAL POINT IS Κ\KAPPAΚ
SUPERCOMPACT	COMPACTNESS EXTENDS FAR BEYOND ITS OWN CARDINALITY
HUGE	STRONGEST COMMONLY CONSIDERED CARDINAL TYPE

THESE AXIOMS LEAD TO VAST IMPLICATIONS IN TOPOLOGY, CATEGORY THEORY AND LOGIC.

ALTERNATIVE FOUNDATIONS OF MATHEMATICS

BEYOND ZFC, VARIOUS FOUNDATIONAL SYSTEMS OFFER ALTERNATE PERSPECTIVES:

SYSTEM	KEY IDEA
TYPE THEORY	HIERARCHY OF TYPES PREVENTS PARADOXES
CONSTRUCTIVISM	AVOIDS LAW OF EXCLUDED MIDDLE (INTUITIONISM)
TOPOS THEORY	USES CATEGORY THEORY INSTEAD OF SET THEORY
HOMOTOPY TYPE THEORY (HOTT)	USES SPACES, PATHS, EQUIVALENCE OVER IDENTITY

THESE ARE HIGHLY RELEVANT IN PROOF ASSISTANTS LIKE COQ, AGDA AND LEAN.

REVERSE MATHEMATICS

GOAL:

FIGURE OUT WHAT AXIOMS ARE NECESSARY TO PROVE CERTAIN THEOREMS.

USES SUBSYSTEMS OF SECOND-ORDER ARITHMETIC

SHOWS MANY CLASSICAL THEOREMS EQUIVALENT TO ONE OF FIVE CORE SYSTEMS (THE “BIG FIVE”)

CONNECTS SET THEORY WITH LOGIC, COMPUTABILITY, AND REAL-WORLD MATH.

SET-THEORETIC FORCING IN INTUITIONISTIC LOGIC

SET THEORY TRADITIONALLY LIVES IN CLASSICAL LOGIC, BUT WHAT HAPPENS IN INTUITIONISTIC OR CONSTRUCTIVE CONTEXTS?

FORCING CAN BE ADAPTED TO HEYTING-VALUED MODELS

USED IN KRIPKE MODELS, REALIZABILITY, AND CONSTRUCTIVE SET THEORIES LIKE CZF

ABSOLUTENESS AND SHOENFIELD’S THEOREM

ABSOLUTENESS:

SOME STATEMENTS ARE TRUE IN ALL TRANSITIVE MODELS OF ZF — THEY ARE “ABSOLUTE”.

SHOENFIELD’S ABSOLUTENESS THEOREM:-

Σ₁² STATEMENTS ARE ABSOLUTE BETWEEN ANY TRANSITIVE MODELS CONTAINING THE SAME ORDINALS.

USED IN:

FORCING INDEPENDENCE PROOFS

COMPARISON OF MODELS

SET-THEORETIC PLURALISM AND PHILOSOPHY

QUESTIONS FROM SET THEORY’S PHILOSOPHY:

IS THERE ONE TRUE UNIVERSE OF SETS (V)?

OR ARE THERE MANY EQUALLY VALID UNIVERSES (MULTIVERSE)?

CAN CH HAVE A “RIGHT” ANSWER, OR IS IT FOREVER UNDECIDABLE?

DO LARGE CARDINALS EXIST ONTOLOGICALLY OR ARE THEY SYNTACTIC TOOLS?

PHILOSOPHERS AND LOGICIANS DEBATE BETWEEN:

PLATONISM: SETS EXIST IN A REAL SENSE

FORMALISM: SET THEORY IS JUST SYMBOLIC MANIPULATION

CONSTRUCTIVISM: TRUTH = PROVABILITY

ULTIMATE THEORETICAL EXERCISES

DESCRIBE THE RELATIONSHIP BETWEEN LLL, VVV, AND LARGE CARDINALS.
CAN A MODEL OF ZFC PROVE ITS OWN CONSISTENCY?
SHOW WHY FORCING DOES NOT COLLAPSE CARDINALS UNLESS EXPLICITLY CONSTRUCTED TO DO SO.
COMPARE THE SEMANTICS OF ZFC AND HOTT.
ANALYZE HOW A PROOF ASSISTANT (LIKE LEAN) USES DEPENDENT TYPE THEORY INSTEAD OF ZFC.

SET THEORY – MODERN INTERSECTIONS AND FUTURISTIC EXTENSIONS (PART 7)

SET THEORY IN ARTIFICIAL INTELLIGENCE (AI)

APPLICATIONS:

KNOWLEDGE REPRESENTATION:-

SET-THEORETIC CONSTRUCTS DEFINE ONTOLOGIES AND SEMANTIC HIERARCHIES.

NON-MONOTONIC LOGIC:–

MODELING UNCERTAIN, DYNAMIC KNOWLEDGE BASES.

FORMAL SEMANTICS:-

SETS UNDERLIE LOGIC USED IN NATURAL LANGUAGE PROCESSING (NLP).

SYMBOLIC AI REASONING:-

MANY INFERENCE SYSTEMS OPERATE USING SET ALGEBRA FOUNDATIONS.

SET THEORY IN MACHINE LEARNING & DATA SCIENCE

WHILE MODERN ML IS LARGELY STATISTICAL, SET THEORY APPEARS IN:

CLUSTERING & CLASSIFICATION:–

PARTITIONS, UNIONS, AND INTERSECTIONS OF SETS.

FUZZY SETS:–

DEGREE OF MEMBERSHIP BETWEEN 0 AND 1; USED IN RULE-BASED ML.

POWER SETS & SUBSETS:-

FOR FEATURE SELECTION AND DIMENSIONALITY REDUCTION.

ROUGH SETS:-

MODEL INCOMPLETE/IMPRECISE INFORMATION (USED IN DATA MINING).

QUANTUM SET THEORY

QUANTUM MECHANICS CHALLENGES CLASSICAL LOGIC — SO CAN WE EXTEND SET THEORY?

QUANTUM LOGIC:–

REJECTS THE LAW OF DISTRIBUTIVITY.

QUANTUM SET THEORY EXPLORES SET MODELS WITHIN ORTHOMODULAR LATTICES.

ISHAM–BUTTERFIELD APPROACH:-

USES TOPOS THEORY TO GENERALIZE QUANTUM SETS.

RESULT:–

A RETHINKING OF SET MEMBERSHIP AND TRUTH VALUES IN QUANTUM CONTEXTS.

NEUTROSOPHIC SETS AND LOGIC

DEFINITION:

EACH ELEMENT HAS DEGREES OF:

TRUTH (T)

INDETERMINACY (I)

FALSITY (F)

WHERE: T,I,F⊆[0,1]T, I, F \SUBSETEQ [0,1]T,I,F⊆[0,1], AND NOT NECESSARILY SUMMING TO 1.

USED IN:-

UNCERTAIN REASONING

DECISION THEORY

ADVANCED DATA FUSION

SET THEORY AND BLOCKCHAIN TECHNOLOGY

SET THEORY PROVIDES A LANGUAGE FOR:

SMART CONTRACT LOGIC (E.G., CONDITIONS MODELED VIA SETS)

CONSENSUS MECHANISMS (E.G., VALIDATOR SETS)

MERKLE TREES (SUBSET VERIFICATION)

ACCESS CONTROL MODELS

FORMAL VERIFICATION OF BLOCKCHAIN PROTOCOLS OFTEN RELIES ON SET-THEORETIC LOGIC.

SET THEORY AND TOPOS THEORY

A TOPOS IS A GENERALIZATION OF SET THEORY VIA CATEGORY THEORY.

A TOPOS:

HAS OBJECTS AND MORPHISMS

SUPPORTS LOGIC, QUANTIFIERS, AND TRUTH VALUES

CAN REPLACE ZFC IN CERTAIN FOUNDATIONAL FRAMEWORKS

EXAMPLES:

ETCS (ELEMENTARY THEORY OF THE CATEGORY OF SETS)

SHEAF MODELS FOR INTUITIONISTIC LOGIC

SET THEORY IN COGNITIVE SCIENCE

COGNITIVE MODELS USE SETS TO SIMULATE MENTAL CATEGORIES, RELATIONS, AND HIERARCHIES.

CONCEPTUAL SPACES OFTEN ASSUME SET-THEORETIC STRUCTURES.

SET THEORY UNDERPINS FORMAL ONTOLOGIES IN COMPUTATIONAL LINGUISTICS AND PSYCHOLOGY.

SET THEORY IN GAME THEORY AND DECISION SCIENCE

STRATEGY SPACES:-

DEFINED AS SETS OF MOVES.

PAYOFF FUNCTIONS:-

MAP CARTESIAN PRODUCTS OF STRATEGY SETS TO OUTCOMES.

SET-VALUED FUNCTIONS:-

REPRESENT UNCERTAIN PREFERENCES OR MULTI-CRITERIA DECISIONS.

ALSO USED IN MECHANISM DESIGN AND MULTI-AGENT SYSTEMS.

FORMAL VERIFICATION AND TYPE THEORY

SET THEORY → BASIS OF CLASSICAL VERIFICATION

TYPE THEORY → MODERN ALTERNATIVE IN PROOF ASSISTANTS (COQ, LEAN, AGDA)

SET THEORY AIDS:

VERIFICATION OF HARDWARE/SOFTWARE

LOGIC SYNTHESIS

PROTOCOL VALIDATION

USED IN TOOLS LIKE:

ISABELLE/HOL

TLA+ (TEMPORAL LOGIC OF ACTIONS)

SET THEORY IN METAMATHEMATICS AND SELF-REFERENCE

KEY AREAS:-

LÖB’S THEOREM

DIAGONALIZATION

FIXED-POINT THEOREMS

SET THEORY PLAYS A ROLE IN DEFINING META-SYSTEMS, FORMAL LANGUAGES, AND PROOF TREES — CONNECTING TO GÖDEL NUMBERING, TURING MACHINES, AND RECURSIVE FUNCTION THEORY.

ADVANCED EXPLORATION EXERCISES

MODEL A REAL-WORLD ML CLASSIFICATION PROBLEM USING ROUGH SETS.
DEFINE A SET-THEORETIC VERSION OF BLOCKCHAIN VALIDATOR LOGIC.
CONSTRUCT A TRUTH TABLE USING NEUTROSOPHIC LOGIC.
COMPARE CLASSICAL SET MEMBERSHIP WITH FUZZY AND QUANTUM ANALOGS.
DEFINE A TOPOS AND EXPLORE HOW LOGICAL OPERATIONS CHANGE INSIDE IT.

SET THEORY – VISUAL TOOLS, PROBLEM STRATEGIES & TEACHING EXTENSIONS (PART 8)

VENN DIAGRAMS – VISUALIZING SET RELATIONS

VENN DIAGRAMS ARE POWERFUL FOR UNDERSTANDING SET OPERATIONS LIKE:

UNION:-

A∪BA \CUP BA∪B → AREA COVERED BY BOTH CIRCLES

INTERSECTION:-

A∩BA \CAP BA∩B → OVERLAP OF CIRCLES

DIFFERENCE:-

A−BA – BA−B → PART OF AAA NOT IN BBB

SYMMETRIC DIFFERENCE:-

A△BA \TRIANGLE BA△B → AREA IN AAA OR BBB BUT NOT BOTH

TEACHING TIP:-

USE 2 OR 3-SET DIAGRAMS FOR ILLUSTRATING COMPLEX IDENTITIES AND PROVING DE MORGAN’S LAWS VISUALLY.

SET-THEORETIC IDENTITIES – CHEAT SHEET

HERE’S A QUICK LIST OF FUNDAMENTAL IDENTITIES:

IDENTITY	FORMULA
IDEMPOTENT LAW	A∪A=AA \CUP A = AA∪A=A, A∩A=AA \CAP A = AA∩A=A
COMMUTATIVE LAW	A∪B=B∪AA \CUP B = B \CUP AA∪B=B∪A, ETC.
ASSOCIATIVE LAW	(A∪B)∪C=A∪(B∪C)(A \CUP B) \CUP C = A \CUP (B \CUP C)(A∪B)∪C=A∪(B∪C)
DISTRIBUTIVE LAW	A∪(B∩C)=(A∪B)∩(A∪C)A \CUP (B \CAP C) = (A \CUP B) \CAP (A \CUP C)A∪(B∩C)=(A∪B)∩(A∪C)
DE MORGAN’S LAWS	(A∪B)′=A′∩B′(A \CUP B)’ = A’ \CAP B'(A∪B)′=A′∩B′
DOUBLE COMPLEMENT	(A′)′=A(A’)’ = A(A′)′=A

INCLUDE THESE IN PRINTABLE WORKSHEETS OR INFOGRAPHICS!

PROBLEM-SOLVING TECHNIQUES

COMMON STRATEGIES:-

USE UNIVERSAL SET UUU WHEN CALCULATING COMPLEMENTS.
TRANSLATE WORD PROBLEMS INTO SYMBOLIC NOTATION.
APPLY VENN DIAGRAMS OR TABLE METHODS FOR SURVEY-BASED PROBLEMS.
USE TRUTH TABLES TO VERIFY SET IDENTITIES.
BREAK DOWN COMPLEX EXPRESSIONS USING DISTRIBUTIVE AND DE MORGAN’S LAWS.

WORD PROBLEMS – TEMPLATES

PROBLEM TYPE 1: SURVEY PROBLEMS

EXAMPLE:
IN A GROUP OF 100 STUDENTS:

60 LIKE MATH

40 LIKE SCIENCE

25 LIKE BOTH

FIND: NUMBER OF STUDENTS WHO LIKE ONLY ONE SUBJECT.

SOLUTION:

MATH ONLY = 60 − 25 = 35

SCIENCE ONLY = 40 − 25 = 15

TOTAL ONLY ONE = 35 + 15 = 50

PROBLEM TYPE 2: LOGICAL PROOFS

EXAMPLE:
PROVE A⊆B⇒A∩B=AA \SUBSETEQ B \RIGHTARROW A \CAP B = AA⊆B⇒A∩B=A

PROOF: IF EVERY ELEMENT OF AAA IS IN BBB, THEN ALL ELEMENTS OF A∩BA \CAP BA∩B ARE JUST THOSE OF AAA, SO THEY’RE EQUAL.

CARDINALITY PRACTICE

FINITE SETS:

USE N(A∪B)=N(A)+N(B)−N(A∩B)N(A \CUP B) = N(A) + N(B) – N(A \CAP B)N(A∪B)=N(A)+N(B)−N(A∩B)

EXTEND TO 3 SETS:
N(A∪B∪C)=N(A)+N(B)+N(C)−N(A∩B)−N(B∩C)−N(C∩A)+N(A∩B∩C)N(A \CUP B \CUP C) = N(A) + N(B) + N(C) – N(A \CAP B) – N(B \CAP C) – N(C \CAP A) + N(A \CAP B \CAP C)N(A∪B∪C)=N(A)+N(B)+N(C)−N(A∩B)−N(B∩C)−N(C∩A)+N(A∩B∩C)

INFINITE SETS:-

COMPARE CARDINALITY USING BIJECTIONS

PROVE N∼Z∼Q\MATHBB{N} \SIM \MATHBB{Z} \SIM \MATHBB{Q}N∼Z∼Q

SHOW ∣R∣>∣N∣|\MATHBB{R}| > |\MATHBB{N}|∣R∣>∣N∣

MULTIPLE CHOICE PRACTICE SET (MCQS)

EXAMPLE:

Q: IF A={1,2,3}A = \{1, 2, 3\}A={1,2,3} AND B={2,3,4}B = \{2, 3, 4\}B={2,3,4}, THEN A△B=A \TRIANGLE B =A△B=
A) {1, 4}
B) {2, 3}
C) {1, 2, 3, 4}
D) {1, 2}

ANSWER: A) {1, 4}

✅ GREAT FOR QUIZZES, WORKSHEETS, OR PRACTICE APPS.

VISUAL TEACHING RESOURCES

SET CARDS:–

INTERACTIVE CARDS WITH SET IDENTITIES AND OPERATIONS

MATCHING GAMES:-

MATCH THE VENN DIAGRAM TO THE EQUATION

INTERACTIVE TIMELINE:–

FROM CANTOR TO COHEN

SET PUZZLES:–

“GUESS THE SET FROM CONDITIONS” GAMES

CROSS-LINK WITH ALGEBRA & LOGIC

BOOLEAN ALGEBRA:-

SETS UNDER UNION/INTERSECTION MIRROR BOOLEAN OPERATIONS

PROPOSITIONAL LOGIC:-

SET IDENTITIES ALIGN WITH LOGICAL EQUIVALENCES

GROUP THEORY:-

COSETS AS PARTITIONS, SUBSETS AS SUBGROUPS

REINFORCE INTERDISCIPLINARY UNDERSTANDING!

TEACHING SET THEORY ONLINE

TOOLS YOU CAN USE:

GEOGEBRA:-

VISUAL VENN DIAGRAMS

DESMOS:-

SET GRAPHS AND INEQUALITIES

MATHIGON:-

INTERACTIVE SET THEORY MODULES

LATEX:-

FOR CLEAR FORMULA-BASED BLOGGING OR RENDERING

GAMIFY IT!

SET THEORY GAME IDEAS:

SET QUEST:-

COMPLETE LOGIC-BASED CHALLENGES TO UNLOCK “SET GEMS”

PROOF BATTLES:-

SOLVE IDENTITIES TO DEFEAT AI OPPONENTS

VENN PUZZLE MASTER:-

DRAG SETS TO MATCH TARGET INTERSECTIONS

EDUCATIONAL GAMES = MORE ENGAGEMENT!

SET THEORY – QUIZZES, FLASHCARDS, AND INTERACTIVE LEARNING (PART 9)

MULTIPLE-CHOICE QUESTIONS (MCQS) – BEGINNER LEVEL

Q1: WHICH OF THE FOLLOWING IS A SUBSET OF EVERY SET?
A) {0}\{0\}{0}
B) {∅}\{ \EMPTYSET \}{∅}
C) ∅\EMPTYSET∅
D) UUU

✅ ANSWER: C) ∅\EMPTYSET∅

Q2: IF A={1,2,3}A = \{1, 2, 3\}A={1,2,3} AND B={3,4,5}B = \{3, 4, 5\}B={3,4,5}, WHAT IS A∪BA \CUP BA∪B?
A) {1,2,3}\{1, 2, 3\}{1,2,3}
B) {3}\{3\}{3}
C) {1,2,3,4,5}\{1, 2, 3, 4, 5\}{1,2,3,4,5}
D) {1,2,4,5}\{1, 2, 4, 5\}{1,2,4,5}

✅ ANSWER: C) {1,2,3,4,5}\{1, 2, 3, 4, 5\}{1,2,3,4,5}

CONCEPTUAL FLASHCARDS (FRONT → BACK)

FLASHCARD FRONT	FLASHCARD BACK
POWER SET	THE SET OF ALL SUBSETS OF A SET
A∩BA \CAP BA∩B	INTERSECTION: ELEMENTS IN BOTH A AND B
A−BA – BA−B	DIFFERENCE: ELEMENTS IN A NOT IN B
CARDINALITY OF FINITE SET A	NUMBER OF ELEMENTS IN A
COMPLEMENT OF A IN U	ELEMENTS IN U NOT IN A

THESE CAN BE MADE PRINTABLE OR APP-BASED (QUIZLET/ANKI FORMAT).

TRUE/FALSE QUIZ SECTION

Q1: EVERY INFINITE SET HAS A POWER SET LARGER THAN ITSELF.
✅ TRUE

Q2: A⊆BA \SUBSETEQ BA⊆B IMPLIES B⊆AB \SUBSETEQ AB⊆A.
❌ FALSE

Q3: THE POWER SET OF THE EMPTY SET HAS ONE ELEMENT.
✅ TRUE → P(∅)={∅}\MATHCAL{P}(\EMPTYSET) = \{\EMPTYSET\}P(∅)={∅}

MATCHING EXERCISES (DRAG & DROP STYLE)

MATCH THE SYMBOL TO ITS MEANING:

SYMBOL	MEANING
∪\CUP∪	UNION
∩\CAP∩	INTERSECTION
⊆\SUBSETEQ⊆	SUBSET
∅\EMPTYSET∅	EMPTY SET
P(A)\MATHCAL{P}(A)P(A)	POWER SET OF A

✅ USEFUL FOR MOBILE QUIZZES OR ONLINE LEARNING APPS.

PROBLEM GENERATOR TEMPLATES

USE THESE DYNAMIC TEMPLATES TO AUTO-GENERATE PROBLEMS:

TEMPLATE 1:-

SET OPERATION WORD PROBLEMS

“IN A GROUP OF __ PEOPLE, __ LIKE A, __ LIKE B, AND __ LIKE BOTH. FIND HOW MANY LIKE ONLY A.”

➡️ PLUG RANDOM VALUES FOR DYNAMIC QUIZ GENERATION.

TEMPLATE 2 – PROOF-TYPE:-

“PROVE THAT IF A⊆BA \SUBSETEQ BA⊆B, THEN A∪B=BA \CUP B = BA∪B=B.”

➡️ USE AS WRITTEN OR AUTO-RANDOMIZE SET EXPRESSIONS.

VENN DIAGRAM-BASED INTERACTIVE QUESTIONS

ASK:
“SHADE THE REGION REPRESENTING: A∩B′A \CAP B’A∩B′”
✅ LET STUDENTS DRAG SHADED AREAS ON AN INTERACTIVE VENN CANVAS.

YOU CAN ALSO:

UPLOAD PRESET IMAGES FOR PRINTABLE WORKSHEETS

USE JAVASCRIPT/CANVAS TO ENABLE WEB INTERACTIVITY

FILL-IN-THE-BLANKS QUIZ SECTION

Q1: THE NUMBER OF SUBSETS OF A SET WITH 4 ELEMENTS IS ___.
✅ ANSWER: 16

Q2: THE __ OF SETS AAA AND BBB IS THE SET OF ELEMENTS THAT BELONG TO BOTH.
✅ ANSWER: INTERSECTION

PROGRESSIVE QUIZ LEVELS

YOU CAN ORGANIZE QUIZZES LIKE A GAME OR COURSE MODULE:

LEVEL	FOCUS	EXAMPLE QUESTION TYPE
LEVEL 1	BASIC OPERATIONS & NOTATION	IDENTIFY SUBSET/SUPERSET
LEVEL 2	SET ALGEBRA AND DE MORGAN’S LAWS	SIMPLIFY (A∪B)′∩C(A \CUP B)’ \CAP C(A∪B)′∩C
LEVEL 3	WORD PROBLEMS WITH VENN DIAGRAMS	FIND HOW MANY LIKE ONLY B
LEVEL 4	PROOFS AND LOGICAL REASONING	PROVE OR DISPROVE SET IDENTITIES
LEVEL 5	INFINITY, CARDINALITY, CH, ETC.	COMPARE CARDINALITIES, PROVE COUNTABILITY

PRINTABLE WORKSHEETS FOR PRACTICE

BEGINNER SHEET: UNION, INTERSECTION, SUBSET

INTERMEDIATE SHEET: WORD PROBLEMS, PROOF WRITING

ADVANCED SHEET: POWER SETS, INFINITY, LOGIC

EACH SHEET COULD HAVE ANSWERS ON THE SECOND PAGE!

WOULD YOU LIKE ME TO PREPARE PDF VERSIONS?

BUILD AN INTERACTIVE WEB APP (CONCEPT)

IMAGINE A LEARNING APP THAT INCLUDES:

LESSONS WITH COLLAPSIBLE SECTIONS

FLASHCARDS WITH AUTO-SHUFFLE

QUIZZES WITH INSTANT FEEDBACK

TIMED GAMES FOR SET EXPRESSION SOLVING

PROGRESS TRACKING DASHBOARD

BUILDING A SUBSCRIBER BASE WITH GAMIFIED CONTENT

SET THEORY – REAL-WORLD APPLICATIONS AND INTERDISCIPLINARY CONNECTIONS (PART 10)

SET THEORY IN DATA SCIENCE AND ANALYTICS

FEATURE SETS & TAGGING:

IN MACHINE LEARNING, EACH DATA POINT HAS ATTRIBUTES, WHICH ARE OFTEN MODELED AS SETS OF FEATURES.

EXAMPLE:
USER PROFILE = {“LIKES SPORTS”, “PREMIUM MEMBER”, “AGE_25_34”}

SET OPERATIONS IN ANALYTICS:

USER SEGMENTS ARE OFTEN MODELED AS SETS.

TO FIND OVERLAPS (E.G., USERS WHO CLICKED AD AND PURCHASED), WE USE INTERSECTION.

TO FIND TOTAL REACH, WE USE UNION.

SETS ARE USED IN SQL IN CLAUSES, GROUP-BY OPERATIONS, AND COHORT ANALYSIS.

SET THEORY IN COMPUTER SCIENCE

🧠 LOGIC AND PROGRAMMING:

SET THEORY FORMS THE FOUNDATION OF BOOLEAN LOGIC, WHICH POWERS EVERYTHING FROM CONTROL FLOW TO CIRCUIT DESIGN.

EXAMPLES:

IF X IN A: IS A SET MEMBERSHIP CHECK.

DATA STRUCTURES LIKE HASHSETS, BITSETS, BLOOM FILTERS ARE ALL BASED ON SET OPERATIONS.

DATABASES:

SCHEMA: REPRESENTING TABLE RELATIONS VIA SETS

RELATIONAL ALGEBRA: USES UNIONS, INTERSECTIONS, CARTESIAN PRODUCTS

SET THEORY IN ARTIFICIAL INTELLIGENCE

KNOWLEDGE REPRESENTATION:-

CONCEPTS, FACTS, AND ENTITIES ARE STORED AS SETS OF PROPERTIES.

FUZZY SETS:-

IN FUZZY LOGIC, ELEMENTS BELONG TO SETS WITH DEGREES OF MEMBERSHIP (E.G., 0.7 IN TALL).

SEMANTIC NETWORKS:-

NODES (CONCEPTS) AND EDGES (RELATIONSHIPS) CAN BE MODELED USING SETS AND GRAPHS.

EXAMPLE: WORDNET OR CONCEPTNET USE SETS OF SYNONYMS, HYPERNYMS, ETC.

SET THEORY IN LINGUISTICS AND LANGUAGE MODELING

SEMANTIC FIELDS:-

GROUPING WORDS BY MEANING (E.G., {RUN, SPRINT, JOG}) IS A SET-BASED MODEL.

SYNTAX TREES:-

PARSING USES SETS OF PRODUCTION RULES.

SET INTERSECTIONS HELP IN MEANING DISAMBIGUATION (E.G., WORD SENSES COMMON IN TWO CONTEXTS).

NLP MODELS OFTEN LEARN SETS OF FEATURES PER WORD OR PHRASE.

SET THEORY IN GRAPH THEORY AND NETWORKS

SETS HELP DEFINE:

VERTICES: SET OF NODES VVV

EDGES: SET OF PAIRS E⊆V×VE \SUBSETEQ V \TIMES VE⊆V×V

SUBGRAPHS, DEGREE SETS, NEIGHBORHOODS, AND CLIQUES

APPLICATION IN:

SOCIAL NETWORKS

WEB CRAWLING

TRANSPORTATION SYSTEMS

SET THEORY IN DISCRETE MATHEMATICS

KEY ROLES:

COMBINATORICS:-

POWER SETS, PERMUTATIONS FROM SETS

LOGIC:-

CONNECTIVES ALIGN WITH SET OPERATIONS

FUNCTIONS:-

AS SUBSETS OF CARTESIAN PRODUCTS

SET THEORY IN BIOLOGY AND TAXONOMY

SPECIES CLASSIFICATION:-

ORGANISMS ARE GROUPED INTO SETS BASED ON TRAITS.

GENETICS:-

GENE EXPRESSIONS CAN BE MODELED AS SETS AND THEIR OVERLAPS STUDIED.

ECOSYSTEMS:-

FOOD CHAINS CAN BE SEEN AS HIERARCHIES OF SUBSETS.

SET THEORY IN EDUCATION TECHNOLOGY

USE OF SETS IN:

ADAPTIVE LEARNING:-

MODELING WHAT A STUDENT KNOWS AS A DYNAMIC SET OF CONCEPTS

CURRICULUM DESIGN:-

MAPPING LEARNING OBJECTIVES AS SETS AND SUBSETS

GAMIFICATION:-

REWARD SYSTEMS BASED ON COLLECTING BADGES (SETS)

USED IN EDTECH TOOLS LIKE KHAN ACADEMY, DUOLINGO, COURSERA.

SET THEORY IN PHILOSOPHY AND LOGIC

NAÏVE SET THEORY HELPED FORMALIZE CLASSICAL LOGIC.

RUSSELL’S PARADOX: INFLUENCED THE FOUNDATIONS OF MATHEMATICS.

ZFC (ZERMELO-FRAENKEL + CHOICE): SET THEORY AS A BASIS FOR CONSTRUCTING ALL OF MATH.

SET THEORY = FOUNDATION OF FORMAL SYSTEMS AND LOGIC FRAMEWORKS.

REAL-WORLD PROBLEM SCENARIOS

EXAMPLE 1: MARKETING

“A CAMPAIGN REACHES 3000 PEOPLE ON PLATFORM A, 2000 ON B, AND 500 ON BOTH. HOW MANY UNIQUE PEOPLE WERE REACHED?”

SOLUTION:
USE UNION: A∪B=3000+2000−500=4500A \CUP B = 3000 + 2000 – 500 = 4500A∪B=3000+2000−500=4500

EXAMPLE 2: CYBERSECURITY

“ALERT SYSTEM: SET A = VIRUS THREATS, SET B = PHISHING. WHICH ALERTS ARE BOTH?”

SOLUTION:
USE INTERSECTION A∩BA \CAP BA∩B TO FIND MULTI-VECTOR THREATS.

EXAMPLE 3: HEALTHCARE

“SET OF PATIENTS ALLERGIC TO PENICILLIN: P. SET ALLERGIC TO ASPIRIN: A. WHO CAN SAFELY TAKE BOTH?”

SOLUTION:
U−(P∪A)U – (P \CUP A)U−(P∪A)

SET THEORY – ADVANCED CONCEPTS, PROOFS, AND AXIOMS (PART 11)

AXIOMATIC SET THEORY (ZFC)

THE FOUNDATION OF MODERN SET THEORY IS BASED ON ZERMELO-FRAENKEL SET THEORY WITH THE AXIOM OF CHOICE (ZFC).

CORE AXIOMS OF ZFC:

AXIOM OF EXTENSIONALITY: TWO SETS ARE EQUAL IF THEY HAVE THE SAME ELEMENTS.
AXIOM OF PAIRING: FOR ANY SETS AAA, BBB, THERE IS A SET {A,B}\{A, B\}{A,B}.
AXIOM OF UNION: FOR ANY SET OF SETS, THEIR UNION EXISTS.
AXIOM OF POWER SET: EVERY SET HAS A POWER SET.
AXIOM OF INFINITY: THERE EXISTS A SET WITH INFINITELY MANY ELEMENTS.
AXIOM OF REPLACEMENT: A FUNCTION’S IMAGE OF A SET IS A SET.
AXIOM OF REGULARITY (FOUNDATION): NO SET IS AN ELEMENT OF ITSELF.
AXIOM OF CHOICE (AC): GIVEN ANY COLLECTION OF NON-EMPTY SETS, ONE CAN SELECT ONE ELEMENT FROM EACH.

✅ THESE AXIOMS AVOID PARADOXES LIKE RUSSELL’S PARADOX.

RUSSELL’S PARADOX

SUPPOSE:

LET R={X∣X∉X}R = \{X \MID X \NOTIN X\}R={X∣X∈/X}.
DOES R∈RR \IN RR∈R?

THIS LEADS TO A CONTRADICTION:

IF R∈RR \IN RR∈R, THEN BY DEFINITION R∉RR \NOTIN RR∈/R

IF R∉RR \NOTIN RR∈/R, THEN BY DEFINITION R∈RR \IN RR∈R

THIS CONTRADICTION LED TO THE DEVELOPMENT OF AXIOMATIC SET THEORY.

ORDINAL NUMBERS

ORDINAL NUMBERS GENERALIZE NATURAL NUMBERS TO REPRESENT POSITIONS IN WELL-ORDERED SETS.

EXAMPLES:

0,1,2,…,Ω,Ω+1,…0, 1, 2, \DOTS, \OMEGA, \OMEGA+1, \DOTS0,1,2,…,Ω,Ω+1,…

Ω\OMEGAΩ IS THE FIRST INFINITE ORDINAL

PROPERTIES:

ORDINALS ARE WELL-ORDERED

EVERY NON-EMPTY SET OF ORDINALS HAS A LEAST ELEMENT

CARDINAL NUMBERS

USED TO COMPARE THE SIZES OF SETS, EVEN INFINITE ONES.

KEY IDEAS:

∣A∣=∣B∣|A| = |B|∣A∣=∣B∣ MEANS A BIJECTION EXISTS BETWEEN AAA AND BBB

∣N∣=ℵ0|\MATHBB{N}| = \ALEPH_0∣N∣=ℵ0 (COUNTABLY INFINITE)

∣R∣=C|\MATHBB{R}| = \MATHFRAK{C}∣R∣=C (CONTINUUM, UNCOUNTABLY INFINITE)

✅ CANTOR PROVED: ℵ0<C\ALEPH_0 < \MATHFRAK{C}ℵ0<C

CANTOR’S THEOREM

THE POWER SET OF ANY SET HAS A STRICTLY GREATER CARDINALITY THAN THE SET ITSELF:

∣A∣<∣P(A)∣|A| < |\MATHCAL{P}(A)|∣A∣<∣P(A)∣

PROOF IDEA: NO FUNCTION FROM A→P(A)A \TO \MATHCAL{P}(A)A→P(A) CAN BE SURJECTIVE (DIAGONALIZATION ARGUMENT).

CONTINUUM HYPOTHESIS (CH)

IS THERE A SET WHOSE SIZE IS STRICTLY BETWEEN ℵ0\ALEPH_0ℵ0 AND C\MATHFRAK{C}C?

FORMALLY:

¬∃A:ℵ0<∣A∣<2ℵ0\NEG \EXISTS A: \ALEPH_0 < |A| < 2^{\ALEPH_0}¬∃A:ℵ0<∣A∣<2ℵ0

✅ GÖDEL AND COHEN PROVED CH IS INDEPENDENT OF ZFC — IT CAN NEITHER BE PROVEN NOR DISPROVEN USING STANDARD AXIOMS.

WELL-ORDERING THEOREM

EVERY SET CAN BE WELL-ORDERED (EVERY SUBSET HAS A LEAST ELEMENT)

✅ EQUIVALENT TO THE AXIOM OF CHOICE.

USED IN PROOFS LIKE:

TRANSFINITE INDUCTION

ZORN’S LEMMA

FILTERS AND ULTRAFILTERS

IN TOPOLOGY AND LOGIC, A FILTER IS A SET OF SUBSETS CLOSED UNDER INTERSECTION AND SUPERSETS.

AN ULTRAFILTER IS A MAXIMAL FILTER — FOR ANY SET AAA, EITHER A∈FA \IN \MATHCAL{F}A∈F OR AC∈FA^C \IN \MATHCAL{F}AC∈F, BUT NOT BOTH.

APPLICATIONS IN:-

MODEL THEORY

TOPOLOGY

NON-STANDARD ANALYSIS

GÖDEL’S CONSTRUCTIBLE UNIVERSE (L)

GÖDEL SHOWED:

IF ZFC IS CONSISTENT, THEN ZFC + CH IS CONSISTENT BY CONSTRUCTING THE UNIVERSE L WHERE EVERY SET IS CONSTRUCTIBLE.

IDEA: SETS ARE BUILT IN LEVELS, LIKE L0,L1,…L_0, L_1, \DOTSL0,L1,…, ENSURING DETERMINISM.

LARGE CARDINALS

THESE ARE INFINITE CARDINALS WITH STRONG LOGICAL PROPERTIES.

EXAMPLES:

INACCESSIBLE CARDINALS

MEASURABLE CARDINALS

MAHLO CARDINALS

USED IN:

INNER MODEL THEORY

CONSISTENCY STRENGTH COMPARISONS

BONUS: PROOF CHALLENGE

THEOREM: THE INTERSECTION OF ANY COLLECTION OF SETS IS A SUBSET OF EACH SET IN THE COLLECTION.

PROOF:

LET {AI}I∈I\{A_I\}_{I \IN I}{AI}I∈I BE A COLLECTION OF SETS.
LET X∈⋂I∈IAIX \IN \BIGCAP_{I \IN I} A_IX∈⋂I∈IAI.
THEN, X∈AIX \IN A_IX∈AI FOR ALL I∈II \IN II∈I.
SO, ⋂I∈IAI⊆AJ\BIGCAP_{I \IN I} A_I \SUBSETEQ A_J⋂I∈IAI⊆AJ FOR ANY J∈IJ \IN IJ∈I. ✅