Examination Sheet № 12
| Name: | Surname: | Group: |
| Elementary conjunction and elementary disjunctions. Conjunctive Normal Form and Disjunctive Normal Forms of formulae of Proposition Logic. Prove that for any proposition there is an equivalent proposition which is either DNF, or CNF. | |
| Pascal Identity (give a combinatorial proof). Give a combinatorial proof of C(2, 2n) = 2C(2, n) + n2. | |
| Construct the DNF, CNF and a polynomial for a proposition F(p,q,r) which is true iff (p,q,r) are from {(1, 1, 0), (1, 0, 0), (0, 0, 1), (0, 0, 0)} | |
| How many ways to select 23 ordered elements from a set consisting of 54 elements are there, if repetition is not allowed? | |
| Find a sequence {an : n = 0, 1, …} satisfying the recurrence relation xn + 6 = –6(xn–1 – 1) – 9xn–2, with the initial conditions a0 =1, a1 = –9. |
Examination Sheet № 13
| Name: | Surname: | Group: |
| Language of Predicate Logic. Definition of a term and of a formula of First Order Logic. Examples. | |
| Show in two different ways that C(n, 0) + C(n, 1) + … + C(n, n) = 2n. Show that C(n, 0) – C(n, 1) + … + (–1)kC(n, k) + … + (–1)nC(n, n) = 0. | |
| Construct the DNF, CNF and a polynomial for a proposition F(p,q,r) which is true iff (p,q,r) are from {(1, 1, 0), (1, 0, 0), (0, 0, 1), (0, 0, 0)} | |
| How many ways to select 23 ordered elements from a set consisting of 54 elements are there, if repetition is not allowed? | |
| Find a sequence {an : n = 0, 1, …} satisfying the recurrence relation xn + 6 = –6(xn–1 – 1) – 9xn–2, with the initial conditions a0 =1, a1 = –9. |