Practice Problems (Induction, recursion and Relations ) PDF

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DISCRETE STRUCTURE FOR COMP. SCI. (CS F222)

Practice Problems (Induction, Recursion and Relations)

Induction 1. Prove, by Mathematical Induction, that n(n + 1)(n + 2)(n + 3) is divisible by 24, for all natural numbers n. 2. Prove using mathematical induction that for all n >= 1, a. 1 + 4 + 7 + … + (3n-2) = n(3n-1) / 2 3. Prove, by Mathematical Induction, that

is true for all natural numbers n. 4. Prove that 52n+1 + 22n+1 is divisible by 7 for all n >= 0. 5. Consider the sequence of real numbers defined by the relations x1 = 1 and xn+1 = √ (1 + 2xn) for n ≥ 1. Use the Principle of Mathematical Induction to show that xn < 4 for all n ≥ 1. 6. f1f2 + f2f3 + f3f4 +

… + f2n-1f2n = f2n2 for all n >= 1.

7. Show that n! > 3n for n ≥ 7. 8. Prove the binomial theorem using induction. This states that for all n >= 1, (x + y)n = ∑" #

%$(nCr) xn-r yr

9. Prove that for any positive integer n, a 2nx2n checkerboard C with any one square removed can be tiled using triominoes. (figure shows the four different orientations of triominoes)

10. Prove that every amount of postage that is at least 12 cents can be made from 4-cent and 5-cent stamps. 11. In the parlour game Nim, there are two players and two piles of matches. At each turn, a player removes some (non-zero) number of matches from one of the piles. The player who removes the last match wins. Prove that if the two piles contain the same number of matches at the start of the game, then the second player can always win. 12. Prove that every positive integer n, n≥2, can be expressed as the product of one or more prime numbers. 13. Suppose that p0 = 1 and px = αpx+1 for all x = 1, 2, ... Prove by mathematical induction that pn = 1/αn for n = 0, 1, 2,. ... 14. Prove the correctness of insertion sort (given below) using mathematical induction.

15. Consider the famous Fibonacci sequence {xn}, defined by the relations x1 = 1, x2 = 1, and xn = xn−1 + xn−2 for n ≥ 3. Use an extended Principle of Mathematical Induction in order to show that for n ≥ 1,

Recursion 1. A computer system considers a string of decimal digits a valid code-word if it contains an even number of 0 digits. For instance, 1230407869 is valid, whereas 120987045608 is not valid. Let a_n be the number of valid n-digit code-words. Find a recurrence relation for a_n. 2. Find a recurrence relation for the number of bit strings of length n that contain three consecutive 0’s. 3. Solve the following recurrence relations (i) 𝑎" = 5𝑎")* − 4𝑎")- .,...𝑎% = 1, 𝑎* = 0 (ii) 𝑎" = 3𝑛𝑎")* ., 𝑎% = 2 (iii) 𝑎" = .2𝑎")* + 1 4. Find a recurrence relation with initial condition(s) satisfied by the following sequences. Assume a0 is the first term of the sequence. (i) an = 2n. (ii) an = 3n - 1.

5. Consider a savings plan in which $10 is deposited per month, and a 6% / year interest rate given with payments made every month. If Pn represents the amount in the account after n months, find a recurrence relation for Pn. 6. Give the recursive definition with initial condition for the function f(n) = 5n + 2, n = 1,2,3, .... 7. Consider the following recursive function f(x; y) s.t. f(x; 0) = 0; for all x, and f(x; y) = f(x; y - 1) + x, where x; y are non-negative integers. What does f(x; y) calculate? 8. Let T(n) be the number of comparisons required to find the minimum and maximum integers from a list of n positive integers. Which of the following value of T(n) is correct? (A) T(n) = T(n - 1) + 1 (B) T(n) = T(n - 1) + 2 (C) T(n) = T(n/2) + 1 (D) T(n) = T(n/2) + 2 9. Write a code snippet to find gcd of 2 numbers a and b using recursion (a>b). 10. Tower of Hanoi is a mathematical puzzle where we have three rods and n disks. The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules: a. Only one disk can be moved at a time. b. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack i.e. a disk can only be moved if it is the uppermost disk on a stack. c. No disk may be placed on top of a smaller disk. Solve the problem using recursion. 11. Given a “2 x n” board and tiles of size “2 x 1”, count the number of ways to tile the given board using the 2 x 1 tiles. A tile can either be placed horizontally i.e., as a 1 x 2 tile or vertically i.e., as 2 x 1 tile. 12. Write a recursive code snippet to implement the following sorting algorithm

The above sorting algorithm is known as insertion sort. 13. What does fun() do in general i.e. which mathematical operation does it implement ( +, -, *, / , xy, &, |, ~, etc.)?

14. Predict the output of fun(99). What does the following fun() do in general?

Hint: This function is known as McCarthy 91 function.

15. What does the following recursive algorithm do?

16. Find the number of ways that a given integer, X can be expressed as the sum of the Nth powers of unique, natural numbers. For example, if X=13 and N=2 and, we have to find all combinations of unique squares adding up to 13. The only solution is 2^2 + 3^2. 17. In k-partition problem, we need to partition an array of positive integers into k disjoint subsets that all have equal sum and they completely covers the set.

18. Given an integer, write a recursive function that returns true if the given number is palindrome, else false. For example, 12321 is palindrome, but 1451 is not palindrome.

19. What does the following recursive algorithm do?

20. Given a string S, find count of all contiguous substrings starting and ending with same character.

21. What does the following recursive algorithm do? Also give the output.

22. Given an area of N X M. You have infinite number of tiles of size 2i X 2i, where i = 0, 1,

2… so on. The task is to find minimum number of tiles required to fill the given area with tiles.

Relations 1. Which of the following relations are reflexive, symmetric and antisymmetric? R1 = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} R2 = {(1,1),(1,2),(2,1)} R3 = {(1,1),(1,2), (1,4),(2,1),(2,2),(3,3),(4,1),(4,4)} R4 = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)} R5 = {(1,1),(1,2), (1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} R1 = {(3,4)}

2. Let A and B be the set of all students and the set of all courses at a school, respectively. Suppose that R1 consists of all ordered pairs (a, b), where a is a student who has taken course b, and R2 consists of all ordered pairs (a, b), where a is a student who requires course b to graduate. What are the relations a. R1 ∪ R2 b. R1 ∩ R2 c. R1 ⊕ R2 d. R1 − R2 e. R2 − R1? 3. Let R be the relation {(1, 2), (1, 3), (2, 3), (2, 4), (3, 1)}, and let S be the relation {(2, 1), (3, 1), (3, 2), (4, 2)}. Find S ◦ R. 4. Let R be the relation on the set of people consisting of pairs (a, b), where a is a parent of b. Let S be the relation on the set of people consisting of pairs (a, b), where a and b are siblings (brothers or sisters). What are S◦R and R◦S? 5. Suppose that the relation R on a set is represented by the matrix

Is R reflexive, symmetric, and/or antisymmetric? 6. Suppose that the relations R1 and R2 on a set A are represented by the matrices

What are the matrices representing R1 ∪ R2 and R1 ∩ R2? 7. Find the matrix representing the relation S ◦R, where the matrices representing R and S are:

8. Find the matrix representing the relation R2, where the matrix representing R is:

9. Let R be the relation on N given by xRy iff x divides y. Determine which of the following properties applies to each relation. (i) Reflexive (ii) Irreflexive (iii) Symmetric (iv) Antisymmetric (v) Asymmetric (vi) Transitive

10. Determine whether the relations for the directed graphs shown in the following figures are reflexive, symmetric, antisymmetric, and/or transitive.

MCQs 11. The number of relations on an “n” element set that are symmetric is: (A) .2"

8

(B).2"(")*)

;(;=) 8

(D).3

12. Let R be the relation on R given by xRy if and only if x < y + 1. (A) Reflexive, but not symmetric and not transitive. (B) Reflexive, symmetric and not transitive. (C) Not Reflexive, not symmetric and not transitive. (D) Reflexive, but not symmetric and transitive. 13. Which of the relations on the given sets are antisymmetric? (S1) A = {1, 2, 3, 4, 5}, R = {(1, 3), (1, 1), (2, 4), (3, 2), (5, 4), (4, 2)} (S2) set of real numbers xRy iff x2 = y2. (A) Only S1 (B) Only S2 (C) Both S1 and S2

(D) Nither S1 nor S2

14. Which relation R is not transitive? (A) {(1, 1), (2, 2)} (C) {(1, 2), (2, 1), (1, 1), (2,2)}

(B) {(1, 2), (2,3), (1, 3)} (D) {(1, 2), (3, 3)}

15. Which of the following is equivalence relation? (A) v. Then R is Equivalence relation, Total Order relation, Partial Order relation? 44. Consider the set S = {a, b, c, d}. Consider the following 4 partitions 𝜋* , 𝜋- , 𝜋L , 𝜋M , on S : 𝜋* = {𝑎𝑏𝑐𝑑 } , 𝜋- {𝑎𝑏, 𝑐𝑑 }, 𝜋L = {𝑎𝑏𝑐, 𝑑}, 𝜋M = {𝑎,𝑏, 𝑐, 𝑑}

Let ≺ be the partial order on the set of partitions S’ = (𝜋*, 𝜋- , 𝜋L , 𝜋M) defined as follows: 𝜋T ≺ 𝜋U if and only if 𝜋T refines 𝜋U . The poset diagram for (S’, ≺) is

45. Draw the hasse diagram of relation R = {(1,1), (2,2), (3,3), (4,4), (5,5), (3,1), (1,4), (5,1), (3,4), (5,4)} 46. In a partially ordered set, a chain is a totally ordered subset. For example, in the set 1, 2, 3, 4, 5, 6, the divisibility relation is a partial order and 1, 2, 4 and 1, 3, 6 are chains. What is the longest chain on the set {1, 2 . . . n} using the divisibility relation? 47. What is the longest chain on the power set of a set A with |A| = n with the ⊆ relation? 48. Given Poset ({3,5,9,15,24,45},/). a) Find the maximal elements. b) Find the minimal elements. c) Is there a greatest element? d) Is there a least element? e) Find all upper bounds of {3,5}. f) Find the least upper bound of {3,5}, if it exists. g) Find all lower bounds of {15,45}. h) Find the greatest lower bound of {15,45}, if it exists. 49. Consider R with usual order ≤ : a). Find lub{x ∊ R : x < 73} b). Find lub{x ∊ R : x2 < 73}

c). Is [R; ≤] a lattice?

50. For the elements x, y, z in a poset, show that if lub[x,y] = a and lub[a,z] = b, then lub[x,y,z] = b. 51. List all the binary relations on the set {0, 1}. 52. A company makes four kinds of products. Each product has a size code, a weight code, and a shape code. The following table shows these codes: Size Code Weight Code Shape Code #1 42 27 42 #2 27 38 13 #3 13 12 27 #4 42 38 38 Find which of the three codes is a primary key. If none of the three codes is a primary key, explain why. 53. If X =(Fran Williams, 617885197, MTH 202, 248B West), find the projections P1,3(X) and P1,2,4(X). 54. R and S are relations on {a, b, c, d}, where R = {(a, b), (a, d), (b, c), (c, c), (d, a)} and S = {(a, c), (b, d), (d, a)}. Find the following combination of relations. i) R2, ii) R3, . iii) S2. iv) S3. v) R : S. vi) S : R. 55. Calculate R-1, where R is the relation on {1, 2, 3, 4} such that a R b means |a − b|...