MATH1090: Set theory
Rigorous course notes on logic, sets, and number construction, written in short linked sections with careful proofs and examples.
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Chapter 1
Logic
Reasoning tools for statements, connectives, and quantifiers.
1.1 Propositional logic
Learn how mathematicians treat statements, connectives, and validity.
1.2 Truth tables and equivalence
Build truth tables and use them to test equivalence, tautologies, and contradictions.
1.3 Quantifiers and negation
Translate quantifiers carefully and negate them without losing meaning.
Chapter 2
Sets and relations
Basic set language, functions, and relations.
2.1 Sets and set operations
Understand membership, subsets, and the main set operations by working with concrete examples.
2.2 Functions and relations
Connect sets to functions and relations, then read injective, surjective, and relational language with confidence.
Chapter 3
Numbers by construction
How natural numbers, integers, and rationals are built, and where Q still falls short.
3.1 Natural numbers and Peano axioms
Meet natural numbers through the Peano viewpoint and learn what the successor operation is really doing.
3.2 Induction and recursive arithmetic
Use induction as a proof pattern and read recursive formulas for + and · without losing the base case.
3.3 Integers from equivalence classes
Build the integers from pairs of natural numbers and read each equivalence class as one signed number.
3.4 Rationals and well-defined operations
Define rational numbers as equivalence classes and check that the usual formulas do not depend on the representative you pick.
3.5 Gaps in Q and why sqrt(2) is not rational
See why Q still has holes by looking at the irrational number sqrt(2) and the set of rationals below it.
Note collections
MATH1090: Set theory
Rigorous course notes on logic, sets, and number construction, written in short linked sections with careful proofs and examples.
Chapter 1
Logic
Reasoning tools for statements, connectives, and quantifiers.
Chapter 2
Sets and relations
Basic set language, functions, and relations.
Chapter 3
Numbers by construction
How natural numbers, integers, and rationals are built, and where Q still falls short.