MA132 Foundations

0. Lecture notes .ps, .pdf

Corrections to lecture notes:

1. Page 11, Principle of Induction II, Property 2: should say "0,1,...,n in S"

2. Page 19, line 3 of Subsection 4.1: should say "We know that hcf(365,750)=5"

3. Page 38, Line 3 of Section 7: should say "... x is the variable or indeterminate, and n is greater than or equal to 0".

4. Page 52, Proof of 7.18: Should say "If P(alpha_1)=...=P(alpha_n)=0 with ..."

5. Page 57, Example 8.2.2; instead of "... let F(n) (for "factors")..." it should say "... let D(n) (for "divisors")...".

6. Page 67, line 2 of proof of 8.18: Instead of "... such that i(x)=y....", it should say "... such that p(x)=y. ..."

7. Page 90, line 6 of proof of 9.24: Instead of "... such that b=g_2h_1....", it should say " ... such that b=g_2h_2. ...".

Calendar

Week 1: Natural numbers and induction Pages 1-10

Week 2 Fundamental Theorem of Arithmetic; hcf and lcm; Euclidean algorithm; rational and irrational numbers Pages 11-20

Week 3 Rationals and irrationals; decimal expansions; approximation by rationals Pages 20-28

Week 4 Set Theory; truth tables; polynomials and polynomial long division Pages 29-39

Week 5 Irreducibility, hcf and lcm of polynomials; Euclidean algorithm for polynomials; binomial theorem; Pages 40-48

Week 6 Pascal's triangle; Remainder theorem, (Fundamental theorem of algebra), algebraic numbers; counting and mappings Pages 49-58

Week 7 Composition of mappings; Bijections, injections, surjections; cartesian product and graphs Pages 59-65

Week 8 Countability; uncountability;binary operations Pages 65-77

Week 9 Groups; arithmetic modulo n; isomorphism Pages 78-89

Week 10 Equivalence relations; Lagrange's Theorem; permutation groups Pages 90-99

Lecture sequence

Exercises and solutions   Exams

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