Some chapters of the study book, such as chapters 10 and 11 go beyond the traditional course of discrete mathematics. AnB The complement of the setB with respect to the setA, or the difference of the setsA and B.
Fundamentals of Mathematical Logic
Problems
Mark with the proposition: "I am hungry", in B "it is now three o'clock". and inC "it's time to eat dinner.". So it's either not three o'clock now, or it's not time for dinner.
Formulate a proposition contrapositive to the following
Point to the statement: "the sun is shining," by B "it's hot outside". and by C "it is snowing.".
Formulate a proposition contrapositive to the following
Formulate a proposition contrapositive to the following
Check by compilation of truth tables whether the following propositions are equivalent
Find which of the following propositions are identically false
Captain of the King’s musketeers, De Treville, received three letters from his subordinates
Using quantifiers, write the statement: "the limit of the function f(x) at the point x0 is not equal to A". Using quantifiers, write the statement: "The function f(x) is not continuous at the point x0".
By the mathematical induction method, prove the proposition
Write the statement using quantifiers: "the limit of the numeric sequence{an} is not equal to". Write the statement using quantifiers: "there exists the only value of variablenx for which the predicate P(x) takes the true value.".
By the mathematical induction method, prove the proposition
As is known from the course in mathematical analysis [86], the number is a limit of the numerical sequence if and only if for any positive number ε there exists a number 0 such that for all natural numbers greater than or equal to ton0, the inequality |an−a|< ε is satisfied.
By the mathematical induction method, prove the propositions
Using the method of mathematical induction, prove that for all natural numbers and the number an−1 is divisible by a−1. Using the method of mathematical induction, prove that the following numbers are divisible by 10 for all values n =1,2,.
Using the method of mathematical induction, prove that the sum Sno of the first terms of the arithmetic progression is found by the formula Sn= a1+an. Using the method of mathematical induction, prove that the sum Sno of the first terms of the geometric progression is calculated by the formula Sn =b1(qn−1).
Prove the identity 1
Prove the identity 1
Formulate the principle of mathematical induction with the help of quantifiers
By forward reasoning, prove the truth of the proposition
Using the method “by contradiction,” prove
Prove irrationality of the following numbers
Write the negation of the predicate
Prove the following inequalities for harmonic numbers [35]
Prove the validity of the relation n
Prove the formula for the sum of the Lucas numbers
It is often more convenient to use an equivalent formula of the mathematical induction principle [4,26]
Prove the criterion of divisibility by 3: For the number N to be fully divisible by 3, it is necessary and sufficient that the sum of the digits of its decimal notation is fully divisible by 3. Prove the criterion of divisibility by 9: For the number N to be fully divisible by 9, it is necessary and sufficient that the sum of the digits of its decimal notation be divided by 9.
Answers, Hints, Solutions
Let's write the truth table for the statements on the left side of the obtained equality. Note that each summan in the sum is presented in the form of the difference.
Hint. Proof can be performed by the methods considered in the Exercise 1.65
Assuming validity of the bond k. Present the sum on the left side of the last equality in the form. For this, write down the sum on the left side of the equality in the form. The first form of the mathematical induction principle follows directly from the second one, since.
Considering the value of the third term of the Fibonacci sequence F3=2, let us rewrite the last equality in the form. Let's prove the truth of the predicate P(k+1) by the recurrence relation for the Fibonacci numbers.
Set Theory
Exercises
Express the operation of set-theoretic difference of sets in terms of the operation of complement of a universal set and the intersection of sets. Which musketeer bothers you the most?" Each respondent named at least one musketeer out of the four: d'Artagnan, Atos, Portos, or Aramis. The only question was: "Which poets' poems do you know by heart?" The results of the questioning are shown in Table 2.3 .
Find the number received by each element of the setA×Bin the summation in accordance with the scheme of the previous exercise's solution. Prove that the kth power of the set of natural numbers Nkfor alk∈N is a countable set.
Answers, Hints, Solutions
Since the number of elements in the set Ais|A| =2, so the cardinality of the power set|P(A)| =4. By representing the defining property of the setBin in the form x2−2x+y20⇔(x−1)2+y21, we conclude that the set Bin includes all points in the plane that lie inside the circle of radius R2=1 centered at the point (1 ,0 ). Use the identity from the previous exercise and the distribution of the set intersection operation with respect to the symmetric difference operation(A B)∩C= (A∩C)(B∩C)[65].
That is why, according to the method of mathematical induction, we obtain the validity of the inclusion and exclusion formula for an arbitrary number of groups. The one-to-one correspondence between the elements of the sets (a,b) and (c,d) is specified by the linear function f(x)= d−c.
Relations and Functions
Problems
Print the set of ordered pairs and draw a directed graph of the relation between the elements of the sets {1,2,3} and {a,b,c,d} defined by the elements of the sets {1, 2,3} and {a,b,c ,d} that defines them. For each of the following relations on the set of natural numbers N, describe the ordered pairs belonging to the relations:. For each of the following relations on the set of non-negative integers Z0= {0} ∪N, describe the ordered pairs belonging to the relations:.
Determine which of the following relations on the set of people is reflexive, symmetric, or transitive:
Find which of the following relations on the set of people are reflexive, sym- metric, or transitive
Find which of the following relations on the set of people are reflexive, sym- metric, or transitive
Find closures with respect to reflexive, symmetric, and transitive properties of the relation
Find closures with respect to reflexive, symmetric, and transitive properties of the relation
Draw Hasse diagram for the following partially ordered set
Draw Hasse diagram for the following partially ordered set
Draw Hasse diagram for the following partially ordered sets
Let R be the relation "...sister..." and S the relation "...husband..." on the set of all people.
Check the fulfillment of Dilworth’s theorem for each of the following partially ordered sets
Check the fulfillment of Dilworth’s theorem for each of the following partially ordered sets
For each of the following functions that operate on ZinZ, indicate whether they are injections, surjections, or bijections:.
Draw graphs of the functions
Draw graphs of the functions
Draw graphs of the functions
Construct the graphs of the following functions
Find the values of the expressions
Find the values of the expressions
How many applicants should enter the same faculty so that
How many times three dice should be thrown so that one could state for sure
How many movies should be in the video rental shop so that the names of some four movies will begin with the same letter?
Answers, Hints, Solutions
It is not symmetric, since the sum+2 is an odd number only if it is odd, andm can be either even or odd, but the sum+2n will be odd only if it is odd. The transitivity of this relation follows from the fact that the sums +2m and +2l are odd only if the sum is odd, and the sum +2l turns out to be odd. It is not reflexive, since it is even only for even, and not transitive, since for odd, even the land, productnmandmlare even, andnlis odd.
We construct Hasse diagram taking into account the information about the direct ancestors of the elements of setA, mentioned in Table 3.2. The logical matrix MR−1 of the relation R−1 is connected to MR by means of the transposition operation: MR−1 =(MR)T.
Hint. Use the result of Exercise 3.43
Associate each element ∈ A, 1i |A| with C(i)—the maximal chain from the chains ending on this element. Relationship of item (1) is not a function, since the element b∈ Ain it is associated with two elements of the setB: 2 and 4. This implication is false since the values of the function gon several arguments a1,a2∈Z coincide, for example, g(6)= g(1)=3.
The left and right sides of the correlation −x = −x coincide for all real x∈R. The equality −x = −xis proved similarly. This is why the set of possible values of the sum of the points on the die B = {3,.
Combinatorics
Problems
There is no limit to the number of students for work placement at the same company. Find the number of square matrices of rows and columns with elements from the set. Find the number of three-digit numbers whose decimal notation contains at least one digit "5".
Find the number of four-digit numbers whose decimal notation contains at least one digit "9". After generalizing the results of Exercises 4.10 and 4.11, find the number of zero-digit numbers whose decimal notation contains at least one of the digits.
Having generalized the results of Exercises 4.10 and 4.11, find the number of n-digit numbers, whose decimal notation contains at least one of the figures
How many different compositions of the organizing committee can be made if it must include at least two students from 11th "A". class and at least one student of the 11th "D" class. How many ways are there to do this if there are at least three programmers to choose from. How many ways are there to do this, if it is necessary to ensure the participation of representatives of all three groups of employees.
How many ways to do this if students from all years except the first are to participate? What are the ways to do this if the first and fourth year students cannot act simultaneously in the film.
Prove Pascal’s 3 identity
Prove the following relation of binomial coefficients to Fibonacci numbers [4], defined in exercise 1.86:. kun0,a is the integral part of the number; see exercise 3.79. Pascal's triangle is formed as follows: The first row contains one element, the second - two, the third - three, etc.; its apex and both sides are formed by elements equal to one; every other element is equal to the sum of the two nearest elements in the row above. It is customary to number the elements of Pascal's triangle starting from zero, and that is why sen=0,1,2,.
Prove that the product of the numbers marked “•” is equal to the product of the numbers marked “◦” [23]. Prove that all numbers, except the first and the last, form the sequence of Pascal's triangle with number 2n, where=2,3,4,.
Answers, Hints, Solutions
The number of choice options will decrease compared to the previous case. The number of outcomes of the event A1 is equal to the number of (6,2)- combinations without repetition, i.e., the binomial coefficient C(6,2). It is clear that the number of solutions of the original equation in non-negative integers is C(a+n−1,n−1), orC(a+n−1,a).
The value C(n,k) determines the number of methods for selecting different elements of the set of n elements (eg set A). The connection number of the graph (G) is the minimum number of connected subgraphs of the graph G.
Directed Graphs
A graph is referred to as a Hamilton2graph if it contains a Hamilton cycle (circuit), i.e. a cycle that goes through every vertex of the graph, and only once. If it is possible to construct a regular-arc coloring for a given graph, the graph is called k-colorable. The out-degree of a vertexv of the digraphD is the number of buerd+(v) of the digraph, leaving from, and the in-degree of this vertex is the number of buerd-(v) entering it.
Connected is a digraph from which a connected graph is obtained after leaving the direction of the arcs. A directed graph is said to be strongly connected if for every pair of verticesu, v∈V there exists a path fromutov.
Problems
Which of the graphs shown in Figure 5.2 can be subgraphs of the graph G. Draw a graph⎡ G whose adjacency matrix has the form:. Which of the graphs in Figure 5.2 can be a subgraph of the graph G. What properties does the adjacency matrix of a simple graph have. Find the chromatic number χ: (1) of the entire graphKn;. 2) a graph obtained from a fully-node graph by removing one edge.
Using the pigeon principle, prove that if a tree T has more than one vertex, then at least two vertices have the same degree. Prove that the chromatic number of the forest F satisfies the inequalityχ(F) 2. Draw the following binary trees:.
Answers, Hints, Solutions
