ARCHIVE PAGE

This page is not updated any more.

Study guide for the final
(MATH 6101-001, Fall 2025)
  1. Definitions and notions to remember:
    Families of sets (Definition 3.4.2), infinite unions and intersections (Definition 3.4.3), functions (Definition 4.1.1), special maps, restriction and extension (Definition 4.1.3), image and inverse image (Definition 4.2.1), composing functions (Definition 4.3.1), coordinate function (Definition 4.3.3), left and right inverses (Definition 4.3.6), injections, surjections and bijections (Definition 4.4.2), Relations (Definition 5.1.1), relation class (Definition 5.1.3, I will provide this one, if needs to be used), reflexive, symmetric and transitive relations (Definition 5.1.5), congruence modulo n and operations on congruence classes (Definitions 5.2.1, 5.2.8 and 5.2.10), equivalence relation (Definition 5.3.1), equivalence classes (Definition 5.3.3), quotient of an equivalence relation (Definition 5.3.6), canonical map (Definition 5.3.8), partition (Definition 5.3.10), Peano postulates (Axiom 6.2.1), cardinality (Definition 6.5.1), finite, countable and infinite sets (Definition 6.5.4), smaller or equal sets (Definition 6.5.9), cardinality of a finite set as a natural number (Definition 6.6.1).

  2. Exercises you should be able to perform:
    Write a set in set notation, using a defining property, or by listing its elements, provide left inverses and right inverses to functions, prove statements by induction.

  3. Statements you should be able to prove:
    Properties of the subset relation (Lemma 3.2.4), laws on union and intersection (Theorem 3.3.3), laws involving set difference (Theorem 3.3.8), properties of the Cartesian product (Theorem 3.3.12), Properties of image and inverse image (Theorem 4.2.4), associative and identity laws for composing functions (Lemma 4.3.5), every left inverse equals every right inverse (Lemma 4.3.7), composing injections and surjections (Lemma 4.4.4), you may cancel injective maps on the left and surjective maps on the right (relevant half of Theorem 4.4.6), congruence modulo n is an equivalence relation (Lemma 5.2.3), there are n congruence classes modulo n (Theorem 5.2.4), congruence modulo n is compatible with addition and multiplication (Lemma 5.2.11), equivalence classes form a partition of the underlying set (Theorem 5.3.4), summation formulas for the arithmetic and geometric sequences (from your notes, see also Proposition 6.3.3), having the same cardinality is an equivalence (Lemma 6.5.2), the set of natural numbers is infinite (Lemma 6.5.5), no set has the same cardinality as its power set (Theorem 6.5.7), Schroeder-Bernstein Theorem (Theorem 6.5.10), a subset of a countable set is countable (Theorem 6.6.7, only basic ideas as we did in class), the union of a countable family of countable sets is countable (Theorem 6.6.9), the Cartesian product of finitely many countable sets is countable (Theorem 6.6.10), every infinite set has a countably infinite subset (Theorem 6.6.11), the set of rational numbers is countably infinite (Theorem 6.7.1), the set of real numbers is uncountable (Theorem 6.7.3), any finite Cartesian power of the set of real numbers has the same cardinality as the set of real numbers (Theorem 6.7.5).

  4. Statements whose proof techniques you should be able to reproduce to prove similar statements:
    Summation formula for an arithmetic sequence (Proposition 6.3.3), inequalities like Proposition 6.3.7, every positive integer is one, a prime, or a product of finitely many primes (Theorem 6.3.10), closed form formula for the Fibonacci numbers (formula (6.4.1), I would provide it, you would need to prove it by induction).

  5. Statements you should be able to state (without proof):
    Properties of infinite unions and intersections (Theorem 3.4.5), one sided inverses, injectivity and surjectivity (Theorem 4.4.5), Division Algorithm (Theorem A.5), equivalence in terms of equivalence classes (Corollary 5.3.5), if a function is constant on the equivalence classes, it factors through the canonical map (Lemma 5.3.9), definition by recursion (Theorem 6.2.3), basic properties of addition, multiplication, and the "greater than" relation (Theorem 6.2.4), Well-Ordering Principle (Theorem 6.2.5), Principle of Mathematical Induction and variants (Theorems 6.3.1, 6.3.6, 6.3.8, 6.3.9), properties of functions on finite sets (Theorem 6.3.11/ parts 1, 2 and 3), the power set of the natural numbers is uncountable (Corollary 6.5.8), "smaller or equal" is antisymmetric on sets (Schroeder-Bernstein Theorem, Theorem 6.5.10), Trichotomy Law for Sets (Theorem 6.5.13), two finite sets have the same cardinality exactly when they are associated with the same natural number (Lemma 6.6.2 and Corollary 6.6.3), rules on cardinalities of finite sets (Theorem 6.6.5), a set is infinite iff it contains an infinite subset (Corollary 6.6.6), equivalent descriptions of a countable set (Theorem 6.6.8), the set of algebraic numbers is countably infinite (Theorem 6.7.2), the set if irrational numbers has the same cardinality as the set of real numbers (Theorem 6.7.4).

  6. What to expect
    The exam will be closed book. The above guide is meant to help with the mandatory part. For the optional part prepare as if it was another midterm. The mandatory part will be as long as the midterm, the optional part will have only about 5 questions. On the mandatory part, some questions may ask you to state and prove a theorem from the list above, others may be like the exercises from your homework assignments. Even if a statement is listed "without proof" above, you must remember the proof of those parts of it that were on a homework assignment!