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Set Theory & Logic

Sets, relations, functions, cardinality, proof and the foundations of mathematics.

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Introductory Sets and operations Membership, subsets, union, intersection, complement, and Venn diagrams. {1,2,3,4} union {3,4,5,6} Core Relations, functions and equivalence Relations as sets of pairs; reflexive, symmetric, transitive; functions and bijections. truth table of p implies q Advanced Cardinality and infinity Countable and uncountable sets, Cantor's diagonal argument, the continuum. sum of 1/2^n for n = 0 to oo Core Logic and methods of proof Connectives, quantifiers, direct proof, contrapositive, contradiction, induction. truth table of (p implies q) iff (not p or q)

What mathematics is built on. Sets and the operations on them, relations and functions defined precisely, the sizes of infinity, and the logic that turns an argument into a proof.

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Sets

Basic Set ConceptsSubsetsUnderstanding Venn DiagramsSet Operations with Two SetsSet Operations with Three SetsSet theorySet (mathematics)SubsetUnion (set theory)Intersection (set theory)Complement (set theory)Power setCartesian productEmpty setVenn diagram

How a proof is made

Mathematical proofDirect proofProof by contradictionProof by contrapositiveMathematical inductionConstructive proofProof by exhaustionCounterexampleConjectureTheoremAxiomLemma (mathematics)CorollaryWell-ordering principleReductio ad absurdumEuclid's theoremCantor's diagonal argument

Relations and functions

Binary relationEquivalence relationPartial orderFunction (mathematics)BijectionInjective functionSurjective functionInverse functionComposite function

Infinity

Cardinal numberOrdinal numberCountable setUncountable setContinuum hypothesisAxiom of choiceZermelo–Fraenkel set theoryRussell's paradoxWell-orderTransfinite induction

Logic

Mathematical logicPropositional calculusFirst-order logicTruth tableLogical connectiveQuantifier (logic)Predicate (mathematical logic)Gödel's incompleteness theoremsModel theoryProof theoryComputability theoryTuring machineHalting problemPeano axiomsBoolean algebra

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Are some infinities bigger than others?

Yes. The integers and the rationals can be listed; the real numbers cannot (Cantor's diagonal argument), so there are strictly more reals than integers.

What is the difference between a relation and a function?

A relation pairs inputs with outputs freely; a function is a relation in which every input gets exactly one output.

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