Tutorial to the course Mathematical Structures in the summer term 2025/26
In every lecture, but the last one, 4 HWs will be posted. Solutions, or attempts at them, should be sent by e-mail to the lecturer (M. Klazar or A. Pultr) by the end of the next Tuesday, best in the pdf format. In the next tutorial on Wednesday, correct solutions will be discussed. To get credits for the tutorial, you need to (attempt to) solve at least 1/2 of the HW problems.

Lecture 1, Feb 18, 2026. HW1 What is the intersection of an empty set? HW2 Let (a, b) = {{a}, {a, b}} be (Kuratowski's) ordered pair. Prove that (a, b) = (c, d) if and only if a =c and b = d. Here = is the set-theoretic equality. HW3 Let X be a finite set. Find the formula for the number of elements |P(X)| in the powerset P(X) of X, in terms of |X|. Let X and Y be any sets and f be a bijection from X to Y. Define, in terms of f, a bijection from P(X) to P(Y). HW4 Prove that if R is a relation on a set X, then the inclusion-wise smallest transitive relation S on X that contains R is the infinite union S = R U (R o R) U (R o R o R) U ...