Content:
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Propositions
- Statements: propositions and predicates.
- Logical connectives (negation, conjunction, disjunction,
implication, and equivalence) and truth tables.
- Propositional forms.
Correct usage of logical symbols. Equivalence of propositional forms.
- Attaching emotions to logical connectives (logical conjunction
vs. colloquial {\sl and, but, though, nevertheless}, etc.).
- Conditional and biconditional sentences, different linguistic expressions associated with
conditionals and biconditionals (``sufficient", ``necessary", ``sufficient and necessary", ``whenever", ``if and only if", etc.).
- Difference between implication in mathematics and causation in language/everyday life.
- Logical identities and how to prove them. Tautology and contradiction, de Morgan's laws, the law of excluded middle and the law of consistency.
- Constructing of useful denials of propositional forms.
- The contrapositive, the converse, and the inverse of a conditional statement.
- Disjunctive and conjunctive normalizations of a propositional form.
- Quantifiers
- Predicates (propositions involving variables).
- Quantifiers and various linguistic expressions associated with them.
- Translating propositions formulated in a colloquial English into symbolic forms and the other way around.
- Analyzing and constructing propositions involving several
quantifiers. Free and dummy variables.
- When quantifiers commute and when they don't.
- Constructing useful denials of propositional forms and quantified sentences.
- Logical structure of a definition. Mathematical definitions as
biconditional sentences with a single free variable. An agreement about
conditional colloquial expressions in definitions.
- Logical structure of a mathematical theorem. Impossibility of
free variables in formulations of theorems.
- Proofs
- Structure of a mathematical theory: basic objects,
propositions: axioms and theorems.
- The role of proofs.
- Distinguishing the formulation (statement)
of a theorem and its proof and the difference between motivation and proof.
- Examples and counterexamples. When and why examples can't replace a proof.
- Proofs of different types: direct proof, proof by contraposition, proof by contradiction, proof by exhaustion.
- Typical logical mistakes, like affirming the consequent or denying the antecedent.
- Principle of mathematical induction in various forms (induction, strong induction, well-ordering principle).
- Sets
- Basic notions of set theory: set and its elements, empty set, subset, intersection, union, difference and complement.
- Relations between logical and set-theoretical operations, like negation and complement, conjunction and intersection etc.
- Set-theoretical identities and how to prove them.
- Definition and properties of a power set.
- Definition and properties of the Cartesian product of sets.
- Definition and examples of a binary relation on a set.
- Properties associated with a binary relation on a set: reflexivity, irreflexivity, symmetry, antisymmetry, transitivity.
- Strict and non-strict partial orders. Linear orders.
- Equivalence relations and partitions of a set.
Equivalence classes and the quotient set.
One-to-one correspondence between equivalence relations on a set
and partitions of the set.
- Examples of equivalence relations from modular arithmetics.
- Maps
- Basic terminology related to maps: the domain, codomain, and range of a map; the image and preimage of a set; the graph of a map.
- Special maps: identity, constant, inclusion. Submaps and restrictions of a map. Projection map and quotient map.
- Definition and properties of characteristic function of a set.
- Metric on a set.
- Definition an properties of a composition of maps.
- Injection, surjection and bijection.
- Definition and properties of inverse map. Equivalence of invertibility and bijectivity.
- Basic examples of functions and their inverse: exponential and logarithmic, tangent and arctangent, etc.
- Cardinality
- Definition of equipotent sets and cardinality of a set.
- Examples of equipotent sets: N and Z, R
and (0,1).
- Addition and multiplication of cardinal numbers. Properties of operations: commutativity, associativity, distributivity.
- Cardinal numbers of the empty set and a singleton.
- Definition and elementary properties of finite sets.
- Pigeon hole principle, its reformulations and corollaries.
- One-to-one correspondence between natural numbers and cardinalities of finite sets.
- Counting principles for finite sets (addition, multiplication, inclusion-exclusion).
- Notions of denumerable, countable and uncountable sets.
- Basic facts about arithmetics of denumerable sets.
- Cantor's theorem about uncountability of the set of all real
numbers.
- Continuum hypothesis.
- Cantor's theorem about cardinalities of a set and its power set.
- Cantor-Schroeder-Bernstein theorem.
- Ordering of cardinal numbers.
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