MATH1090: Set theory
Build precise mathematical reasoning from logic and sets, then construct the number systems, understand completeness and limits, and compare infinite sets. Begin with propositions and work toward proofs that depend on clearly stated assumptions.
1.1 Propositional logic
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Chapter 1
Logic
Reasoning tools for statements, connectives, and quantifiers.
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1.1 Propositional logic
Learn how mathematicians treat statements, connectives, and validity.
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1.2 Truth tables and equivalence
Build truth tables and use them to test equivalence, tautologies, and contradictions.
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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.
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2.1 Sets and set operations
Understand membership, subsets, and the main set operations by working with concrete examples.
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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.
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3.1 Natural numbers and Peano axioms
Meet natural numbers through the Peano viewpoint and learn what the successor operation is really doing.
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3.2 Induction and recursive arithmetic
Use induction as a proof pattern and read recursive formulas for + and · without losing the base case.
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3.3 Integers from equivalence classes
Build the integers from pairs of natural numbers and read each equivalence class as one signed number.
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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.
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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.
Chapter 4
Order and completeness
Total order, bounds, supremum and infimum, and the completeness gap between Q and R.
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4.1 Total orders and ordered fields
Separate total order from partial order, then see how the familiar order on Z and Q works together with field operations.
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4.2 Upper bounds, supremum, and infimum
Distinguish maxima and minima from upper and lower bounds, then learn how supremum and infimum capture the correct extremal language.
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4.3 Completeness and gaps in Q
Define completeness precisely and use the set of rationals below sqrt(2) to see why Q still has genuine gaps.
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4.4 Axioms for the reals and first approximations
Treat the reals as the target complete ordered field, then use decimal approximations to motivate the first construction idea.
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4.5 Dedekind cuts and the embedding of Q
Define a real number as a left/right split of Q, see why rational cuts embed Q faithfully, and understand how order and arithmetic are rebuilt on cuts.
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4.6 Decimal expansions and irrational numbers
Turn infinite decimal intuition into a cut, then use sqrt(2) to see how irrational numbers live inside the completed number system.
Chapter 5
Sequences and first limits
Sequences, Cauchy convergence, and the first delta-epsilon treatment of function limits.
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5.1 Sequences and epsilon-N limits
Treat a sequence as a function on N, then learn the epsilon-N definition of convergence through concrete examples.
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5.2 Cauchy sequences and another model of the reals
See why Cauchy sequences capture internal convergence, then sketch how equivalence classes of rational Cauchy sequences give another model of R.
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5.3 Delta-epsilon limits, limit laws, and continuity
Move from sequence limits to function limits, learn the delta-epsilon definition, and organize the first toolkit of failure tests, limit laws, and continuity.
Chapter 6
Big sets
Cardinality, countability, Cantor's theorem, choice principles, intervals, Cantor set, density, and well-ordering.
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6.1 Cardinality, countability, and cardinal inequalities
Compare finite and infinite set sizes by bijections, injections, and explicit countable enumerations.
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6.2 Cantor's theorem, continuum, and choice
Use Cantor's theorem to prove that power sets are strictly larger, then place continuum and choice principles in context.
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6.3 Intervals, Cantor set, density, and well-ordering
Study intervals, the Cantor set, density, and well-ordering as different ways of measuring size and order.
Chapter 7
Sets with structure
Binary operations and the first algebraic structures built on top of sets.
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7.1 Binary operations, monoids, and groups
Move from bare sets to binary operations, then separate monoids from groups by their algebraic laws.