The first midterm will take place in the 7th week of instruction and the second midterm will be given in the 11th week of instruction, unless announced otherwise.

Each midterm (to be completed in 40 minutes) consists of four problems with
up to 5 points per problem. To obtain **"zápočet"**, a student needs to
earn at least 7 points from the first and 8 points from the second midterm.
There will be an opportunity to re-take one of these tests offered at the
beginning of examination session, but experience shows that one should not
rely on it too much, try to get it right the first time.

If you miss a midterm for a documented reason beyond you control (sickness, earthquake, alien abduction etc.), you will be allowed to take it at a different time. You should contact you instructor as soon as possible, preferably even before the midterm is written (if possible). This is not counted as a re-take, it is the first take, just at different time.

**First midterm** tests knowledge of relations, divisibility, and
induction.

• First problem: Some (easy) proof by induction.

practice problems

• Second problem: You will be given a particular relation and asked to
decide (and justify your answers) whether this relation satisfies some
property (symmetry, transitivity etc.).

practice problems

• Third problem: A question related to ordering (Hasse diagram,
linearization, minima maxima and such) or equivalence (equivalence
classes).

practice problems

• Fourth problem: You will be asked to prove some theoretical property
involving relations or divisibility. Typical examples: A proof that symmetry
passes from *R* to *R* ^{-1}, or a proof that
when *a* divides *b*, then *a* divides *b*^{2}.

**Second midterm** tests knowledge of calculations modulo and congruency,
and recurrence equations.

• First problem: You will be asked to simplify a given expression
modulo some given *n*, little Fermat's theorem should come handy.

practice problems

• Second problem: You will be asked to find all solutions of
*ax* = *b**Z*_{n}, one can also meet the same equation solved as
congruency modulo *n* or a diophantine equation directly.
A system of equations requiring the Chinese remainder theorem is extremely
unlikely, keep it in store for the final exam.

practice problems

• Third problem: Homogeneous recurrence equation.

practice problems

• Fourth problem: One can expect some proof, this time involving not just relations and divisibility, but also congruency modulo.

**Rules:** You can use a simple calculator, it is not allowed to use
smart calculators, cellphones, other people's brains and other subversive
tools.