Content:

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, wellordering principle).
 Sets
 Basic notions of set theory: set and its elements, empty set, subset, intersection, union, difference and complement.
 Relations between logical and settheoretical operations, like negation and complement, conjunction and intersection etc.
 Settheoretical 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 nonstrict partial orders. Linear orders.
 Equivalence relations and partitions of a set.
Equivalence classes and the quotient set.
Onetoone 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.
 Onetoone correspondence between natural numbers and cardinalities of finite sets.
 Counting principles for finite sets (addition, multiplication, inclusionexclusion).
 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.
 CantorSchroederBernstein theorem.
 Ordering of cardinal numbers.
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If you have a physical, psychological, medical, or learning disability that
may impact your course work, please contact SASC (631) 6326748 or
http://studentaffairs.stonybrook.edu/dss/. They will determine with you
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website:
http://www.stonybrook.edu/ehs/fire/disabilities/asp
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For more comprehensive information on academic integrity, including
categories of academic dishonesty, please refer to the academic judiciary
website at
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Critical incident management:
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