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This is a final exam for a university-level course on discrete mathematics, covering topics such as mathematical induction, recurrence relations, big o notation, generating functions, set theory, graph theory, and boolean algebra. The exam consists of 23 questions, each worth 10 points, and allows the use of a calculator and a single 8.5x11 page of notes.
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1. Prove by mathematical induction that 2 n^ > n^2 , n ≥ 5. 2. For n
1, let an be the number of bitstrings of length n which do not contain three consecutive 0’s. (Thus for example a 1 = 2, a 2 = 4, a 3 = 7.) Find a recurrence relation for the sequence {an}. (Do not solve it!) (Hint: See if your solution gives the right value for a 4 .)
5. Give a closed form generating function for the sequence 2, 0, 6, 0, 18, 0, 54, 0, 162, 0,
…. (This means: 2 + 6x^2 + 18 x^4 + 54x^6 + … is not an acceptable answer.) (Hint:
use the identity
1 − x
= 1 + x + x^2 + x^3 + ... .)
6. Let A, B, C, D be finite sets. Write down the inclusion-exclusion formula for
| A ∪ B ∪ C ∪ D |.
7. Let A be a set with 3 elements.
(a) (5 points) How many binary relations are there on the set A? (Hint: the answer is one of 1, 2^2 , 2^3 , 2^4 , … , 2^512 .) (b) How many equivalence relations are there on the set A?
8. Define the relation R on the set of real numbers as follows: For every two real numbers x, y, xRy if and only if the number xy is rational. Prove that R is an equivalence relation, or prove that R is not an equivalence relation.
11. (a) (5 points) Circle the correct answer. Let G be the following graph
A G has a Hamilton cycle. B G has a Hamilton path, but no Hamilton cycle. C G does not have a Hamilton path. (b) (5 points) Circle the correct answer. Let H be the following graph.
A H has a Hamilton cycle. B H has a Hamilton path, but no Hamilton cycle. C H does not have a Hamilton path.
12. (a) (5 points) Draw a simple planar graph, with six vertices, in which each vertex has degree 3. (b) (5 points) Draw a simple non-planar graph, with six vertices, in which each vertex has degree 3.
13. Using a binary tree, make up a binary prefix code for the letters a,b,c,d,e. (There are many different correct answers to this question.) 14. Find the postfix form of the expression (( x + y ) ↑ 2) + (( x − 4) / 3).
17. Prove the Boolean identity x + xy = x. 18. Find the sum-of-products expansion of the function F(x,y,z) = (x+y)z.
19. Define the Boolean operator | (the NAND operator) by 1|1 = 0 and 1|0 = 0|1 = 0|0 = 1. Show that xy = (x|y)|(y|x). 20. (a) (5 points) Circle the correct answer. What is the coefficient of x^8 in the power
series
(1 − x )(1− 2 x )
(b) (5 points) Circle the correct answer. Let G be a simple connected planar graph. Assume that for some planar embedding of G (drawing of G in the plane with no edge-crossings) there are 5 regions, and every region (including the infinite region) has exactly 6 edges in its boundary. (That is, every region has degree 6.) Then the number of vertices of G is:
A 10 B 12 C 30 D 9 E cannot be determined from the given information.
23. Prove that x + xy + xyz + xyzw = x for all possible values of the Boolean variables x,y,z,w. (Hint: There is a very short proof.)
