Solved: 1. Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends. 2. Show that if there are 30 students in a class, then at least two have last names that begin with the same letter.

Define a function fm: N x N ->N as follows: fm(n, k) =k if 0 ≤ n < m, and fm(n, k) =fm(n-m, k+1) otherwise. Describe in terms of a single well-known arithmetic operation what fm(n, 0) is computing.

Consider the statement: The art show was enjoyable but the room was hot. a) Use a variable to represent each basic statement in the given statement. b) Use the variables and logical operators to represent the statement in symbolic for…

Show that A=B A = {1,2,3} B ={n|n∈Z+ and n^2<10}

Let A = {2,3,4} , B = {6,8,10} , and C = {a,b} True or False? (i) 4R6 (ii) 4R8 (iii) (3,8) ∈ R (iv) (2,10) ∈ R (v) (4,12) ∈ R

There are 18 mathematics majors and 325 computer science majors at a college. a) In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major? b) In how many ways can …

Q 4: Find the Number of Mathematics students in a university taking at-least one of the language Mandarin, English and Japan Give the following data: 65 study Mandarin 45 study English 42 study Japanese 20 Study Mandarin and English …

Q. Construct the truth tables for (p & q here are not above statement) a) p V ~q b) ~p V ~q c) q -->p d) ~(p-->q)-->r

how propositional logic can be used in Boolean search? Give detail answer with an example of web page searching.

The generating function of the sequence {1,2,3...n..} is (1-z)².

6. Use truth tables to determine whether the argument form is valid. [5 Marks] a. There is an undeclared variable or there is a syntax error in the ﬁrst ﬁve lines. b. If there is a syntax error in the ﬁrst ﬁve lines, then there is a m…

∀x∃y(x + y = 2 ∧ 2x − y =1

A palindrome number is a number that is read similarly backwards. How many possible 5-digit palindromic numbers are there? A. 608 B. 648 C. 688 D. 728

A keyboard has its keys removed. If the last line of the letters were jumbled, how many ways can I rearrange it so that consecutive letters are beside each other? A.120 B. 144 C. 156 D.196

Five different colored flags can be hung in a row to make a coded signal. How many signals can be made if a signal consists of the display of one or more flags? A. 11 B. 13 C. 15 D. 17

Make negation of the following statements. I4) All students have taken a course in mathematics. 15) Some students are intelligent, but not hardworking

In an engineering class there are 15 students taking chemistry,20 students taking mathematics. Of these students,10 are taking both chemistry and mathematics.How many students are taking either chemistry or mathematics? Expert's answer

Alice decides to set up an RSA public key encryption using the two primes p= 31 and p= 41 and the encryption key e= 11.You must show all calculations, including MOD-calculations using the division algorithm! Bob decides to send the me…

1. Discuss two real world binary problems in two different fields using applications of Boolean Algebra.

Write the following sets in the roster/tabular form: a. G = {x : x ∈ N, 5 < x < 12} b. H = {x : x is a multiple of 3 and x < 21} c. I = {x : x is perfect cube 27 < x < 216} d. J = {x : x = 5n - 3,n ∈ W, and n < 3}

prove with inductions that n^2+n divisible by 2

Prove that (1 * 2) + (2 * 3) + (3 * 4) + (4 * 5) + ....+ n (n + 1) = n (n + 1) (n + 2) / 3 for n positive integers

In a survey, 2000 people were queried if they read India Today or Business Times. It was found 1200 read India Today, 900 read Business Times and 400 read both. How many read at least one magazine

Suppose that the statement p→ ¬q is false. Find all combinations of truth values of r and s for which (¬q→r)∧(¬p∨s) is true.

Let p and q be the propositions ”Swimming at the Sarıyer shore is allowed” and ”Sharks have been spotted near the shore”, respectively. Express each of these compound propositions as an English sentence. (a) ¬p ∨ q (b) p ⇒ ¬q (c) p ⇔ …

Determine the truth values of each proposition below. (a) 1 + 1 = 3 if and only if 2 + 2 = 3. (b) If 1 + 1 = 2 or 1 + 1 = 3, then 2 + 2 = 3 and 2 + 2 = 4. (c) If squirrels play badminton, then cats can’t fly.

If p is true, q, r, and s are false, find the truth value of the proposition. ¬(¬𝑝 ∨ 𝑞) ∨ ¬((𝑝 ∧ ¬𝑟) → ¬𝑠)

There are 7 groups in a picnic who has brought their own lunch box, and then the 7 lunch box are exchanged within those groups. Determine the number of ways that they can exchange the lunch box such that none of them can get their own.

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of a particular suit are chosen

There are 7 groups in a picnic who have brought their own lunch box, and then the 7-lunch boxes are exchanged within those groups. Determine the number of ways that they can exchange the lunch box such that none of them can get their …

Draw the Hasse diagram for (i) Pi = {1, 2, 3, 4, 12} and < is a relation such that x < y if x divides y.

Let P = {1,2,3,4} and R = {(1,1), (2,1), (1,2), (2,2), (3,2), (3,4), (4,3), (4,4)} which is a relation on P. Represent this relation as a directed graph. Check whether this relation is an equivalence relation or not.

For discrete structures there are n exams to check and there are k graders. To guarantee a high quality of grading, every exam may be checked by any number of graders (but always at least by one grader). This means that summed all tog…

An orientation of a graph G = (V,E) is any directed graph G'= (V,E) arising by replacing each edge {u, v} belonging to E, by the directed edge (u, v) or by the directed edge (v, u). Show that for every planar graph there is an orienta…

Draw the De Bruijn graph for 2-tuples with digits 0,1,2, and 3. Make sure to add matching labels to all vertices and edges.

A group of 21 students participates in a discrete mathematics competition. There are Q questions that have to be answered. For each question exactly 4 students are assigned to work together, while the others take a short break. The qu…

Find the sum of product expansion of the Boolean function f(x,y,z) = (x+z)y

For discrete structures there are n exams to check and there are k graders. To guarantee a high quality of grading every exam may be checked by any number of graders (but always at least by one grader). This means that summed all toge…

Let ρ be a relation on a set A. Define ρ −1 = { | ∈ ρ}. Also for two relations ρ, σ on A, define the composite relation ρ ◦ σ as (a, c) ∈ ρ ◦ σ if and only if there exists b ∈ A such that ∈ ρ and ∈ σ. Prove the following assertions (a…

the number of combinations of five objects taken two at a time is?

Show that the propositions pi, p2, p3, and p4 can be shown to be equivalent by showing that p1 ↔ p4, P2 ↔ P3, and p1 ↔ p3.

Show that ~ (p → q) and p ∧~q are logically equivalent. (Hint: you can use a truth table to prove it or you apply De Morgan law to show the ~(p → q) is p ∧~q.

The Mathclub, VIT-AP wants to conduct a group event for its members. So the club president has to fix the group size the event. When he tries to fix the size to be 5 members in each group, 4 members are left; when he tries to fix the …

Make a truth table for the given expression. 12. (~p∧q) ∨ (p∧~q) 13. (p∧~q) ∨ r 14. ~[(p∧~q) ∨ ~p] 15. (p∨q) ∧ ~(~q∧r) 16. (~p∧q) ∨ (~p∨q) 17. ~[(p∨q) ∧ ~q]

Show that among 2000 student in a school, at least six were born on the same day of the leap year

Using propositional-formula and logical connectives, convert these sentences into symbolic form. No doctors are enthusiastic; You are enthusiastic. Therefore, you are not a doctor Dictionaries are useful; Useful books a…

translate the statement "The product of any two positive integers is always positive"into logical expression(using Nested Quantifiers)

Check whether the relation defined by{(a,b)}| a is cousin of b}defined on the sets of all human being is an equivalence or not?

He relation (a,b) such that "a" and "b" have the same age ,is defined on the sets of all people.Is it equivalent relation?

Indian Cricket team was doing catching practice .The fielding coach informed then that they will throw ball to 4 other players done this ,but others did not throw any ball to catch.how many players received the ball including first pl…

If we have two graph G = (V, E) and G0 = (V, E0 ) on the same set V of vertices we call the union G ∪ G0 the graph (V, E ∪ E0 ). Next we have two claim about the chromatic number χ(G ∪ G0 ) of G ∪ G0 .Prove that χ(G∪G0 ) ≤ χ(G)∗χ(G0 )…

An orientation of a graph G = (V, E) is any directed graph G0 = (V, E0 ) arising by replacing each edge {u, v} ∈ E by the directed edge (u, v) or by the directed edge (v, u). Show that for every planar graph there is an orientation su…

Are these system specifications consistent? ”The file system is not locked only if new messages will be queued. If the file system is not locked, then the system is functioning normally, and conversely. For new messages are not queued…

USING MATHEMATICAL INDUCTION P(n)=1+22+2n+........+2n= 2n+1 - 1

A town has two type of people Knave always speak lie and knative always speak truth, you meet a group of six people A, B, C, D, E and F they communicate with you as following, A Said: All are the knative . B Said: Atleast three ar…

A binary operation € on three variables A, B and C define as return true if at least two true return us true then prove that OR operation on three variable is equivalent to which of the following ? A€(B€C) 2) A€(B€¬C) 3) A€(…

How many Combinations of bit strings length 9 have: a) exactly three 0s? b) at least seven 1s?

how many three letter words can be formed by the word letter EQUATION

In a class of 200 students,70 offered physics,90 chemistry,100 mathematics while 24 did not offer any of the three subject,23 student offered physics and chemistry,41 chemistry and mathematics while 8 offered all the three subjects.Ho…

d)Out of 300 students taking discrete mathematics ,60 take coffee, 27take cocoa,36 take tea,17 take tea only,47 take chocolate only,7 take chocolate and cocoa,3 take chocolate, tea and cocoa,20 take cocoa only ,2 take tea, coffee and …

Construct the switching network of the given compound statements. Then, construct another switching network of the equivalent statement by applying the laws of logic: 1. p v (~p v q) v (p v ~q) ] ^ ~q 2. (p → q) ^ (p ↔ q)

Simplify the following expressions using laws of logic and put what law of logic did you use or apply. p v ~(~p --> q) [(p --> q)^ ~q] --> ~p [(p v q) ^ (p --> ~r) ^ r ] --> q (p v ~q) ^ (p v q) 5. ~[p --> ~(p ^ q)]

Let S be the following statement. “If it is raining, then the ground is wet.” Translate the S into symbol State in English the converse of S State in English the contrapositive of S

Simplify the following expressions using laws of logic: p v ~(~p --> q) [(p --> q)^ ~q] --> ~p [(p v q) ^ (p --> ~r) ^ r ] --> q (p v ~q) ^ (p v q) 5. ~[p --> ~(p ^ q)]

Construct two different assignments that are functions. The functions should be represented by means of ordered pairs graphically. Construct domain, codomain and range(image) of each function.

Writing set into a form of {x | P(x) } 1. {2,4,6,8,10} 2. {a,e,i,o,u} 3. {1,8,27,64,125} 4. {-2,-1,0,1,2} If A = {3,7,12} Find: 5. |A| 6. P(A) 7. | P(A) |

Yesterday, Allan, Berto, and Carlo are playing guess that day. Berto and Carlo gives Allan clues and fake clues. Berto said today is Friday, but Carlo said its Saturday. Allan asked what day is it tomorrow? Carlo said its Monday while…

Determine whether each of these functions is a bijection from R to R. f(x)=2x+1 f(x)=x^2+1

Write the set in the form { x | P(x) } 1. {2, 4, 6, 8, 10}

. Identify the contrapositive of the following statement. If x = 5, then x + 8 = 13

Prove the validity of the arguments using rules of inference. { (p→q) ^ [ q → (r ^ s) ] ^ (p ^ w) ^ [~r v (~w v u) ] } → u

List the members of these sets: (a) A={ x∣x is a real number ∋ x2=1} (b) B={ x∣x is a positive integer less than 100 } (c) C={ x∣x is the square of an integer and x <100 } (d) D={ x∣x is an integer ∋ x2=2 }

Show that {1, 2, 3, 4, . . .} and {2, 4, 6, 8, . . .} have the same cardinality. Hint: find a mapping and show it is 1–1 and onto.

1. A. (20p) Give a Big-O estimate that is as good as possible for each of the following functions. i. (n^3 + n)(logn + n) ii. (n + 1000)(logn + 1) B. (15p) If these are complexities of algorithms that solve the same problem, which …

2. Following algorithm calculates the sum of first n nonnegative integers. Prove the algorithm to be correct using mathematical induction. procedure sum(n: nonnegative integer) if n = 0 then return 0 else return n + sum(n - 1) {o…

In set using venn diagram: The are 60 sciences students in ss3, in a secondary school, 35 of whom study chemistry, 30 study technical drawing, 12 out of those students study biology and chemistry but not technical drawing, 10 study ch…

By using mathematical induction prove that (n+1)! > 2^(n+1) for n, where n is a positive integer greater than or equal to 4

Check the validity of the following argument: “If I study then I will not fail in Mathematics. If I do not play Basket ball then I will study. I failed in Mathematics. Therefore I must have played Basketball

2. For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended. Explain your answer. a) Experience with C++ or Python is required. b) Lunch includes soup or salad. c) Publish or perish. d) To e…

By using mathematical induction prove that (n+1)! > 2^(n+1) for n, where n is a positive integer greater than or equal to 4

In a school, students must take at least one of these subjects: Maths, Physics or Chemistry. In a group of 50 students, 7 take all three subjects, 9 take Physics and Chemistry only, 8 take Maths and Physics only and 5 take Maths and C…

solve these non-homogeneous recurrence relation an+ 4an-1 − 5an-2=n+2 where a0=1 & a1=-1

I am having trouble with the following problem. I am unsure on how to even start. Use propositional logic to prove that the following argument are valid: (A⟶B) ⋀ (A⟶(B⟶C)) ⟶ (A⟶C)

A debating team consists of three boys and two girls. Find the number n of ways they can sit in a row if the boys and girls are each to sit together.

Write the sequence of the following explicit formula: 1. sn = (-4)n , 1 <_ n 2. tn = 92 -5n, 1<_n <_5

1. Blue taxi inc. charges $7.50 for the first 4 miles and 1 dollar for each additional miles. The table shows the cost of travelling from 4 to 10 miles. Mileage Cost 4. 7.50 5. 8.50 6. 9.50 7. 10.50 8. 11. 50 9. 12.50. 10. 13.…

Write the first 4 terms of the following formula, start with n = 1. an = 5n nbn = 3!bn = 3n bn = 3n^2 +2n - 6 gn = 1 × 2 × ... n c1 = 2.5, cn = cn-1 + 1.5 d1 = -3, dn = -2dn-1 + 1

Write the formula of the following sequence, identify if explicit or recursive formula or both. 1. 1.3,5.7 2. 1, -1, 1, -1 3. 1, 4, 7, 10, 13, 16 4. 0, 3, 8, 15, 24, 25 5. 0, 2, 0, 2, 0, 2 ... 6.1, 1/2, 1/4, 1/8, 1/16 7. 0, 1, …

Write the sequence of the following explicit formula. 1. sn = (-4)^n, 1 ≤ n 2. bn = 92 - 5n, 1 ≤ n ≤ 5 3. Blue taxi inc. charges $7.50 for the first 4 miles and 1 dollar for each additional miles. The table shows the cost of travel…

Let S denote the set of all companies listed on the Botswana Stock Exchange. Define the following sets as: A = x | x is in the mining sector , B = x | x has an annual turnover exceeding P10 million , C = x | x has a financial ye…

If A is a set, let A¯ denote the complement of A. Show the following (a) (A ∩ B¯) ⊂ A ∩ B (b) A − B = A if and only if B − A = B

Use bit strings to find the union, intersection, A and B compliments of these sets. The bit strings for the sets {2, 3, 6, 7 and 8} and {1, 3, 5, 7, 9} are 0110011100 and 1010101010, respectively.

Express each of the statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the words “It is not the c…

Find the power set of the set {Ø , { Ø }, a, b, c} where a, b and c are distinct elements.

I. p is "x< 50"; q is "x> 40". Express the following compound propositions as English sentences in as natural a way as you can (a) ¬p (b) ¬q (d) p∨q (e) ¬pV q (f) ¬pV ¬q

2-2.7+2.7^2-...+2(-7)^n = (1- (-7)^n+1)/4

If 11 students offer Biology, 13 students offer Maths, 14 students offer Agric, 9 students offer Maths and Biology, 3 students offer Maths and Agric, 7 students offer Maths, Biology and Agric. How many students offer Biology and Agric?

Let p, q, and r be the propositions. p : You get an A on the final exam. q : You do every exercise in this book. r : You get an A in this class. Write these propositions using p, q, and r and logical connectives (including negations).…

Let A, B, and C are not empty sets, then using Venn diagram represent the following: (i) (A ∩ B) ∩ Cc (ii) Ac ∪ (B ∪ C) (iii)(A – B) ∩ C (iv) (A ∩ Bc) ∪ Cc

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