 Assignments (2021) Chapter 1: Problem 1. Let A be the set of (straight) lines on a plane. Consider a relation R such that xRy when lines x and y both contain point (0,0). Explain why R is not an equivalence relation. Problem 2. Show that the lexicographical ordering is transitive. (This is a part of the proof that this relation actually is a strict total order.) Problems 1.7, 1.8 from the lecture notes Deadline: Saturday 23/10 for at least 2 (out of 4) problems. Chapter 2: Problem 2.1: Problem 2.2 from the lecture notes. It is sufficient to consider it for the case of A of 6 elements, or the case of A infinite. Problem 2.2: Consider the poset of the first diagram of problem 2.1 from the lecture notes. Show that it is a lattice. Show that the lattice is not distributive. (Begin with naming the poset's elements.) Alternatively, the same for the poset (P,â‰¤), where P={0,1,a,b,c}, 0â‰¤{a,b,c}â‰¤1, and a,b,c are incomparable. Problem 2.3: Problem 2.12 from the lecture notes, without (3). Deadline: 30/10 for 2.1, 4/11 for 2.2 and 2.3 Chapter 3: problems here (extended on 11/11). Chapter 4: Problem 4.1. Explain what is wrong in this erroneous inductive proof. Begin with making it clear which induction principle is used. (Deadline: as soon as possible:) Problems 4.2, 4.3 Chapter 5: the problems here; deadline - 7/12. Chapter 6: the problems here (except for those possibly solved at a lecture); deadline 15/12 Rules of the game: You are supposed to try to solve all of the above exercises and provide correct solutions to most of them to pass the course. It is permitted to discuss the exercises with others, but you must solve each exercise individually. It is absolutely not allowed to copy or rephrase each others solutions, or to solve the exercises jointly . Solutions handed in after the deadline will be dealt with a low priority, thus graded possibly rather late.

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