MAT 200 Logic, Language and Proof, Fall 2020.

  • Instructor: Oleg Viro, office 5-110 Math Tower, e-mail:
  • Office hours: Monday and Wednesday 11:30am-1:00pm in 5-110 Math Tower or by appointment.
  • Grader: Paul Sweeney Math Tower S-240A
  • Textbook : Peter J. Eccles, An Introduction to Mathematical Reasoning, Cambridge University Press.
  • Class meetings: Monday and Wednesday, 4:25-5:45pm, in Earth & Space 001

    Everyone participating in this class must wear a mask or face covering at all times or have the appropriate documentation for medical exemption. Please contact Student Accessibility Support Center (SASC) at if you need special accommodations. Any student not in compliance with this policy will be asked to leave the class.

  • Homework: will be assigned weekly through Blackboard (Assignments). Your solutions should be submitted to Blackboard. Each submission should contain a single pdf-file. You may use any app which consolidate your pictures in a single pdf-file (for example, CamScan). Submission in a wrong format (multiple files, jpg-format, links to the cloud, etc.) will be accepted but with reduced score. Late submission will be accepted but with reduced score.

    Homework 1 due by 9/9.

    Homework 2 due by 9/9.

    Homework 3 due by 9/16.

  • Quizzes will be given weekly in class.
  • Exams: two midterms (in class) and final exam.
    The Final Exam will be on Wednesday, Dec. 9, 8:30pm-11:00pm online.
  • Grading system: your grade for the course will be based on: homework 15%, quizzes 15%, two midterms 25% each, final exam 20%.

    All your work should be done by you and nobody else. Submitting somebody's else work is a serious violation of university integrity policy and will be treated respectively. See Academic integrity statement below.

    If your in-class work will have significantly lower grade than your combined online work, the instructor reserves the right to arrange a personal Zoom meeting and validate that you are able to reproduce the online work you have submitted. By the results of this meeting, the total grade may be changed.

  • Make-up policy: Make-up examinations are given only for work missed due to unforeseen circumstances beyond the student's control.
  • Content:
    1. 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.
    2. 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.
    3. 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).
    4. 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.
    5. 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.
    6. 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.

    Student Accessibility Support Center (SASC) statement: If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact SASC (631) 632-6748 or They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and SASC. For procedures and information go to the following website:

    Academic integrity statement: Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty are required to report any suspected instance of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website at

    Critical incident management: Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, and/or inhibits students' ability to learn.