Deck 14: Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Question 1

Use mathematical induction to prove that n! ≥ 2n−1 whenever n is a positive integer.

Answer 1

Question 2

Find the sum-of-products expansion of the Boolean function f(x, y, z) that has the value 1 if and only if an odd number of the variables x, y, and z have the value 1.

Answer 2

Question 3

A door lock is opened by pushing a sequence of buttons. Each of the three terms in the combination is entered by pushing either one button or two buttons simultaneously. If there are 5 buttons, how many different combinations are there? (Example: 1-3, 2, 2-4 is a valid combination.)

Answer 3

Question 4

Find the prime factorization of 16,575.

Question 5

(a) Show that the relation is an equivalence relation on the set of real numbers.

Question 6

Prove or disprove that if A and B are sets then A ∩ (A ∪ B) = A.

Question 7

Find a spanning tree for the graph using (a) a depth-first search. (b) a breadth-first search.

Question 8

(a) How many functions are there from a set with three elements to a set with four elements? (b) How many are one-to-one? (c) How many are onto?

Question 9

Suppose that Prove that 5 divides an whenever n is a positive integer.

Question 10

How many positive integers not exceeding 1000 are not divisible by either 8 or 12?

Question 11

A fair coin is flipped until a tail first appears, at which time no more flips are made.

Question 12

Find the set recognized by the following deterministic finite-state machine.

Question 13

How many bit strings of length 10 have at least eight 1's in them?

Question 14

(a) Prove or disprove: If a ≡ b (mod 5), where a and b are integers, then (b) Prove or disprove: If where a and b are integers, then a ≡ b (mod 5).

Question 15

Find a recurrence relation and initial conditions for the number of ways to go up a flight of stairs if stairs can be climbed one, two, or three at a time.

Question 16

Answer the following questions about the graph (a) How many vertices and how many edges are in this graph? (b) Is this graph planar? Justify your answer. (c) Does this graph have an Euler circuit? Does it have an Euler path? Give reasons for your answers. (d) What is the chromatic number of this graph?

Question 17

How many bit string of length 10 have at least one 0 in them?

Question 18

Prove or disprove that

Question 19

Prove or disprove that the fourth power of an odd positive integer always leaves a remainder of 1 when divided by 16.

Question 20

Use mathematical induction to prove that every postage of greater than 5 cents can be formed from 3-cent and 4-cent stamps.

Question 21

(a) Let be a positive integer greater than 2 . Show that the relation consisting of those ordered pairs of integers (a, b) with (mod m) is an equivalence relation.
(b) Describe the equivalence classes of this relation where =4 .

Question 22

How many nonisomorphic unrooted trees are there with four vertices? Draw these trees.

Question 23

(a) Describe the bit strings that are in the regular set represented by (b) Construct a nondeterministic finite-state automaton that recognizes this set.

Question 24

Find the sum-of-products expansion for the Boolean function

Question 25

(a) Does the graph have an Euler circuit? If not, does it have an Euler path?
(b) Does the graph have a Hamilton path?

Question 26

(a) How many functions are there from a set with four elements to a set with three elements? (b) How many of these functions are one-to-one? (c) How many are onto?

Question 27

A thumb tack is tossed until it first lands with its point down, at which time no more tosses are made. On each toss, the probability of the tack's landing point down is (a) What is the probability that exactly five tosses are made? (b) What is the expected number of tosses?

Question 28

How many symmetric relations are there on a set with eight elements?

Question 29

Construct a binary search tree from the words of the sentence This is your discrete mathematics final, using alphabetical order, inserting words in the order they appear in the sentence.