SF2714 Discrete Mathematics and Algebra 7.5 hp KTH, period 2, Fall 2009 Course coordinator: Jakob Jonsson –

Course design 7.5 hp = 5 weeks of full-time work 18 lectures with 90 minutes of theory and problem- solving Course book: Biggs – Discrete Mathematics Examination: ● Two sets of home assignments ● Not mandatory, but give bonus points on exam ● Final exam (December 15)

Lectures 18 lectures, each 90 minutes ● 16 ordinary lectures ● 2 concluding lectures with time for questions Lectures are not mandatory. We discuss the underlying theory and solve problems. The course webpage will be updated with detailed information about each lecture.

Literature Norman L. Biggs, Discrete Mathematics, 2 nd edition, Oxford University Press. Supplementary material ● Will be handed out during lectures or be available for download from the course webpage.

Home assignments Two sets of home assignments The assignments may give up to 8 bonus points on the final exam ● 8 points = 80 % correct on the assignments ● 7 points = 70 % correct on the assignments ● etc.

Final exam 8 problems worth 36 points ● Part I: 3 problems on par with grade E (12 p) ● Part II: 4 harder problems (24 p) You pass with 20 points (including bonus) Examples (Bonus + part I + part II) : = 20 = grade E = 28 = grade C = 36 = grade A

Course content IArithmetic ~ 20 % IICombinatorics~ 15 % IIIGraph Theory~ 15 % IVGroup Theory~ 20 % VFields and Polynomials~ 20 % IVError-Correcting Codes~ 10 %

Arithmetic ● Divisibility – An integer a divides another integer b if b/a is an integer ● Prime numbers – Integers p such that only 1 and p itself divide p ● Modular arithmetic – “Identify” two integers if their difference is a multiple of a given integer m – Arithmetic operations on classes of identified integers rather than on individual integers

Combinatorics ● Binomial numbers – Count the number of subsets of size k of a set of size n ● Principles of counting – Addition principle, multiplication principle, sieve principle ● Set partitions – Subdivisions of a set into subsets ● Permutations – Bijective functions on a set

Graphs ● Basic structures in graphs – Trees and cycles ● Graph colorings – Assign a color to each vertex in a graph such that two adjacent vertices have different colors ● Matchings – Sets of edges in a graph such that no vertex appears in more than one edge

Groups ● Basics of groups – Theory unifying properties of sets with a binary operation – Requirements: the binary operation is associative ; there is an identity ; every element has an inverse ● Examples of groups – Cyclic groups, abelian groups, permutation groups ● Structures within groups – Subgroups, cosets ● Orbits of permutation groups – The concept of permutation cycles lifted to a group level

● Basics of fields – Theory unifying properties of sets with two binary operations: addition and multiplication – Addition and multiplication work as in Q. Examples: – ab = ba ; a(b+c) = ab + ac ; 1/x exists when x is nonzero ● Polynomials over fields – Factorization – Theory analogous to that of the integers ● Finite fields – Fields with finitely many elements – Combine theories of modular arithmetic and polynomials Fields and Polynomials

● Basics of codes – A code is a set of words such that all words are “far apart” – Codewords are often binary words – A few errors in a codeword can be repaired, as all other codewords are “far away” ● Bounds on codes – How large can a code be if we want to correct e errors? ● Linear codes – A code forming a vector space under addition. – Nice application of linear algebra Error-Correcting Codes