| Week # |
Sections |
Topics |
Prerequisite Skills |
Student Learning Outcomes
|
| 1
|
1.1 & 1.2
|
Propositional Logic
|
–
|
- Recognize propositional formulas built from atoms using connectives.
- Correctly interpret propositional formulas using truth tables.
|
|
2
|
1.3 & 1.4
|
- Tautologies and Deductions.
- Quantifiers.
|
|
- Establish whether a propositional formula is a tautology.
- State De Morgan's Laws of logic.
- Recognize conditional tautologies as laws of deduction.
- Express conditionals in disjunctive form.
- Express the negation of a conditional in conjunctive form.
- Identify the direct and contrapositive forms of a conditional.
- Recognize the non-equivalence of a conditional and its converse.
- Recognize a biconditional as the conjunction of a conditional and its converse.
- Identify the domain of interpretation of a quantified statement.
- Correctly interpret quantified statements.
- Correctly negate quantified statements.
|
|
3
|
1.5 & 1.6
|
- Sets.
- Set Operations.
- Introduction to proofs of universal statements in set theory
- Disproving universal statements via counterexamples.
|
- Tautologies and Deductions.
- Quantifiers.
|
- Recognize and interpret set equality and set inclusion.
- Recognize set operations and state their formal definitions.
- Recognize formal proofs as processes of logical deduction of conclusions from assumptions.
- Prove basic universal statements pertaining to set inclusion and set operations.
- Correctly identify false universal statements in set theory and disprove them with appropriate counterexamples.
- Correctly use propositional and quantified tautologies as deductive laws.
|
|
4
|
2.1
|
- Divisibility of integers.
- The Division Algorithm.
|
- Proofs and Counterexamples.
- Propositional Logic.
- Quantifiers.
|
- Recognize the notion of integer divisibility via its formal definition, examples and counterexamples.
- Correctly state and apply the Division Algorithm of integers.
- Prove basic facts pertaining to divisibility and the division algorithm.
|
|
5
|
2.2 & 2.3
|
- Greatest Common Divisor.
- Bèzout's Identity: GCD(a,b) = au + bv for some u,v∊ℤ.
- Coprime integers.
- The Extended Euclidean Algorithm.
|
- Divisibility of integers.
- The Division Algorithm.
|
- Compute the GCD of two integers using the Euclidean algorithm.
- Express the GCD of two integers as a linear combination thereof using the extended Euclidean algorithm.
|
|
6
|
2.5
|
- Primes.
- Euclid's proof of the infinitude of primes by contradiction.
- Euclid's Lemma: for p prime, p|ab implies p∣a or p∣b.
- Unique factorization and the Fundamental Theorem of Arithmetic.
|
- Divisibility of integers.
- The Extended Euclidean Algorithm.
- Greatest Common Divisor.
- Coprime integers.
|
- Define prime numbers and state their basic properties.
- Prove the infinitude of primes.
- Prove Euclid's Lemma using Bèzout's identity.
- Prove uniqueness of prime factorization using Euclid's Lemma.
- Characterize divisibility and GCD of integers in terms of their prime factorizations.
|
|
7
|
3.1–3.3
|
- Arithmetic congruences and basic modular arithmetic.
- Tests of divisibility.
|
- Divisibility of integers.
- The Division Algorithm.
|
- Use arithmetic congruences to interpret the remainder of integer division.
- Use congruences to compute remainders of divisions where the quotient is large or irrelevant.
- Prove basic divisibility criteria by 2, 3, 5, 9 and 11 for number in base 10, using modular arithmetic.
|
|
8
|
3.4
|
- Modular rings ℤₙ.
- Modular fields ℤₚ.
- Fermat's Little Theorem.
|
- Primes.
- Arithmetic congruences and basic modular arithmetic.
|
- Recognize the modular rings ℤₙ as number systems.
- Evaluate sums, differences, negations and products in ℤₙ.
- Identify invertible and non-invertible elements of ℤₙ.
- Find the inverse (when defined) of a given element of ℤₙ.
- Prove that the modular ring ℤₚ is a field if and only if p is prime.
- Correctly state Fermat's Little Theorem, both as a theorem in modular arithmetic modulo a prime p, and as a theorem for the finite field ℤₚ.
- Apply Fermat's Little Theorem to solve arithmetic problems.
|
|
9
|
4.1
|
- Mathematical Induction.
- Inductive proofs.
|
- Basic proofs.
- Tautologies and Deductions.
- Quantifiers.
- Divisibility of integers.
|
- State the Principle of Mathematical Induction (PMI).
- Prove elementary algebraic and arithmetic statements by induction.
- Prove elementary algebraic and arithmetic statements by strong induction.
|
|
10
|
4.2 & 4.3
|
- Recursion.
- The Binomial Theorem (Binomial Expansion Formula).
|
- Mathematical Induction.
- Inductive proofs.
- Factorials.
|
- Recognize recursive definitions of sequences and sets.
- Prove elementary properties of recursively defined sets and sequences (Fibonacci and geometric sequences).
- Recursively construct successive rows of Pascal's triangle.
- Identify the entries in Pascal's Triangle as Binomial Coefficients.
- State and apply the Binomial Expansion Formula.
- Compute individual binomial coefficients using the quotient-of-falling powers formula (n𝑪k) = n(n−1)…(n−k+1)/k!
|
|
11
|
5.1 & 5.2
|
- The rational number system 𝐐.
- The real number system 𝐑.
- Fractional powers and roots of real numbers.
- Rational and irrational numbers. Existence of irrationals.
|
- Divisibility of integers.
- Unique factorization and the Fundamental Theorem of Arithmetic.
- Decimals and decimal expansions.
- Roots and fractional powers of real numbers.
|
- Identify the set 𝐐 of rational numbers as a number system (a field).
- Identify the set 𝐑 of real numbers as a number system (a field extending 𝐐).
- Prove the irrationality of √2 and, more generally, of √p for p prime.
- Prove that fractional powers xm/n of real x>0 are well defined and unique.
- Informally interpret the convergence of decimal expansions as the completeness of 𝐑.
- Informally recognize that the universal existence of roots ⁿ√x and fractional powers xm/n of real numbers x>0 relies on the completeness of 𝐑.
|
|
12
|
8.1–8.4
|
- Introduction to complex numbers and their operations.
- The complex number system 𝐂.
- The complex plane.
|
- The real number system 𝐑.
- Fractional powers and roots of real numbers.
|
- Represent complex numbers algebraically in Cartesian form.
- Represent complex numbers geometrically as points on a plane.
- Carry out arithmetic operations with complex numbers.
- Interpret the geometric meaning of addition, subtraction and complex conjugation.
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