Examination Sheet № 15
| Name: | Surname: | Group: |
| Set. Subset. Equality of two sets. The Power Set. The cardinality of the power set (prove this formula for finite sets by mathematical induction). Cartesian Products. The cardinality of the Cartesian product of two sets. Computer representation of sets. | |
| Binomial coefficient. The binomial theorem. Expand brackets in (a1+a2+…+ak)n. | |
| 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 № 16
| Name: | Surname: | Group: |
| Set Operations. Venn Diagrams of all essential set operations. Set Identities. The cardinality of the union of two sets. Prove by Venn Diagram de Morgan laws. | |
| The Pigeonhole principle. Show that every sequence of n2 + 1 distinct real numbers contains a subsequence of length n + 1 that is either strictly increasing or strictly decreasing. | |
| 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 № 17
| Name: | Surname: | Group: |
| Cardinality of a set. Definitions of |A| = |B| and |A| ≤ |B|. Countable sets. Show that the set of integer and the set of rational numbers are countable. | |
| The Pigeonhole principle. Show that among any n + 1 positive integers not exceeding 2n must be an integer that divides one of the other integers. | |
| 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. |