Solved: In a certain group of 15 students, 5 have graphics calculators and 3 have a computer at home (one student has both). Two of the students drive themselves to college each day and neither of them has a graphics calculator nor a computer at home. A student is selected at random from the group. Let G represent the event “the student has a graphics calculator”, H represent the event “the student has a computer at home”, D represent the event “the student drives to college each day” Represent the information in this question by a Venn diagram. Use the above Venn diagram to answer the questions. (a)Find the probability that the student either drives to college or has a graphics calculator.

multiple-choice test contains 10 questions. There are four possible answers for each question. a) In how many ways can a student answer the questions on the test if the student answers every question? b) In how many ways can a student…

Out of a group of ten residents in a certain county, 3 are Republicans, 5 are Democrats, and 2 are Independents. How many unique partitions of this group of residents are there by political party? 38) You have eight distinct …

Use the binomial theorem to find the coefficient of x^a y^b in the expansion of (2x^3 − 4y^2) 7, where a) a = 9, b = 8. b) a = 8, b = 9. c) a = 0, b = 14. d) a = 12, b = 6. e) a = 18, b…

Find the expansion of a) (x + y)^6 b) (x + y)^4 31) What is the coefficient of x^12 y^13 in the expansion of (x + y)^25? 32) What is the coefficient of x9 in (2 − x)19? 33) What is the coefficient of x^101 y^99 in the e…

bit is either 0 or 1: a byte is a sequence of 8 bits. Find (a) the number of bytes that can be formed (b) the number of bytes that begin with 11 and end with 11 (c) the number of bytes that begin with 11 and do not end with 1…

a) In how many ways can a committee of 5 be chosen from 9 people? (b) How many committees of 5 or more can be chosen from 9 people? (c) In how many ways can a committee of 5 teachers and 4 students be chosen from 9 teac…

4. Let A and B be sets. Prove the commutative laws from Table 1 by showing that a) A ∪ B = B ∪ A. b) A ∩ B = B ∩ A.

How many pairs of dance partners can be selected from a group of 12 women and 20 men?

Let R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)} R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)} is a relation on set A={1,2,3,4} A={1,2,3,4} Suppose a Rn b means that there is a path of length n from a to b Which of the elements are …

Answer the following items. Show your complete answer on a separate sheet of paper. Prove that the following sentences are tautologies. 1. p →p 2. p → (p V q) 3. [p Λ (p → q)] → q 4. p V ~p 5. q → (p V ~p) 6. ~p…

Show that the relation R = ∅ on a nonempty set S is symmetric and transitive but not reflexive.

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Theorem 2:- An (m,n) encoding ∅ corrects K or fewer errors if and only if the minimum distance between encoded words is atleast 2K+1 Proof:- Suppose that ∅ corrects K or fewer errors x is an encoded word of distance 1≤j≤k …

One hundred students were asked whether they had taken course in any of the three areas, CSE, ME and EE. The results were. 45 had taken CSE, 38 had taken ME, 21 had taken EE, 18 had taken CSE and ME, 9 had taken CSE and…

Consider a Boolean expression Ex,y,z=xzv(y^z). Find disjunctive and conjunctive normal form.

For each stall there are 3 ways to exhibit one of three animals. Since there are 10 stalls, by Combinatorial Product Rule, the number of ways of arranging the exhibition is equal to 3\cdot 3\cdot\ldots\cdot 3=3^{10…

There are 10 stalls for animals in an exhibition. Three animals; lion, pussycat and horse are to be exhibited. Animals of each kind are not less than 10 in number. What is the possible number of ways of arranging the exh…

Show that the following logical equivalences hold for the Peirce arrow ↓, where P ↓ Q ≡ ∼(P ∨ Q). a. ∼P ≡ P ↓ P b. P ∨ Q ≡ (P ↓ Q) ↓ (P ↓ Q) c. P ∧ Q ≡ (P ↓ P) ↓ (Q ↓ Q) H d. Write P → Q using Peirce arrows on…

Obtain the Conjunctive Normal Form of (x^y) V (-x^y)

Define and give examples of injective surjective and bijective functions. Check the injectivity and surjectivity of the following function f: NN given by f(x)=x2

A school organized a book fair and in this book fair a book seller is selling his books under the following rules: There are three different packages available. First package contains 2 Islamic books, 2 Science books and 2 Geography…

The argument is p→~q,~r→p,q|–r in true table in mathematical foundations of computer science

How many license plates can be made using either two or three letters followed by either two or three digits?

Show ((p ∨ q) ∧ ¬(¬p ∧ (¬q ∨ ¬r))) ∨ (¬p ∧ ¬q) ∨ (¬p ∧ ¬r) is tautology, by using replacement process.

I come to class whenever there is going to be a quiz this statement is inverse converse and contarpositve?

An engineer designs at least one robot a day for 30 days. If a total of 45 robots have been designed, then show that there must have been a series of consecutive days when exactly 14 robots were designed.

Let an denote the number of surjective (onto) functions f : {1, 2, . . . , n} −→ {1, 2, 3} such that f(1) < f(2). Give a Θ estimate for an.

A committee of 8 is to be formed from 16 men and 10 women. In how many ways can the committee be formed if i) there are no restrictions. ii) there must be 4 men and 4 women. iii) there should be an …

A committee of 8 is to be formed from 16 men and 10 women. In how many ways can the committee be formed if i) there are no restrictions. ii) there must be 4 men and 4 women. iii) there should be an even number of women. …

solve the recurrence t(n)=(t(n/2)^2) assuming t(1)=1

list the ordered pairs in the equivalence relations R induced by these partitions of p { {1} , {3} , { 2,4,5,6} rt he set of { 1,2,3,4,5,6}

Suppose there are 10 male and 6 female professors to teach Discrete mathematics. In how many ways a student can choose Discrete mathematics professor.

Let R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)} R={(1,2),(1,4),(2,1),(2,4),(3,2),(3,4)} is a relation on set A={1,2,3,4} A={1,2,3,4} Suppose a Rn b means that there is a path of length n from a to b Which of the elements are …

The argument is p→~q,~r→p,q|–r in true table in mathematical foundations of computer science

Obtain the Conjunctive Normal Form of (x^y) V (-x^y)

Let a and b be two cardinal numbers. Modify Cantor’s definition of a < b to define a ≤ b. (Hint: Examine what happens if you drop condition (a) from Cantor’s definition of a < b.) 2. Prove that a ≤ a. 3. Prove that if a ≤ b and b ≤ c,…

Draw a simple, undirected graph yourself, the vertices are connected with each other including 8 vertices and 14 edges. Find the shortest path from two arbitrary vertices: a) The weight of each edge is 1. b) Self-weighting for…

Let A, B, C, D denote, respectively, art, biology, chemistry, and drama courses. Find the number N of students in a dormitory given the data: 12 take A, 5 takeAand B, 4 takeB and D, 2 take B, C,D, 20 take B, 7 takeAand C, 3 t…

The following formulas have been abbreviated based on the common abbreviation rules. Follow the steps below and translate the formulas into good English. · Step 1: Re-add the omitted brackets. · Step 2: If necessary, con…

State TRUE or FALSE justifying your answer with proper reason. a. 2𝑛^2 + 1 = 𝑂(𝑛^2 ) b. 𝑛^2 (1 + √𝑛) = 𝑂(𝑛^2 ) c. 𝑛^2 (1 + √𝑛) = 𝑂(𝑛^2 log 𝑛) d. 3𝑛^2 + √𝑛 = 𝑂(𝑛 + 𝑛√𝑛 + √𝑛) e. √𝑛 log 𝑛 = 𝑂(𝑛)

solve the following recurrence relations a. 𝑇(𝑛) = 𝑇( 𝑛/4) + 𝑇( 𝑛/2 ) + 𝑛^2 b. T(n) = T(n/5) + T(4n/5) + n c. 𝑇(𝑛) = 3𝑇( n/4 ) + 𝑐𝑛^2 f. 𝑇(𝑛) = (𝑛/𝑛−5) * 𝑇(𝑛 − 1) + 1 g. 𝑇(𝑛) = 𝑇(log 𝑛) + log 𝑛 h. 𝑇(𝑛) = 𝑇 (𝑛^ 1/ 4) + 1 i. 𝑇(𝑛…

How many positive integers less than 100 is not a factor of 2,3 and 5?

A pet store keeps track of the purchases of customers over a fours hour period. The store manager classifies purchases as containing a dog product, a cat product, a fish product, or product for a different kind of pet. He found! …

For the relation R = {(p,p) ,(q,p),(q,q),(r,r),(r,s),(s,s) ,(s,m) ,(m,m)} 1.Using warshall algorithm find the transitive closure R* of R 2.write matrix representation of R* 3.Check whether the relation of R* is an equiv…

Write the converse, inverse, and contrapositive of the following conditional propositions. (Hint: If applicable, write each conditional proposition in standard form first.) a. Rose may graduate if she has 120 hours of OJT credits. …

How many different plates are there that involve 1, 2 or 3 letters followed by 1, 2, 3 or 4 digits? How many 2 digit or 3-digit numbers can be formed using the digits 1, 3, 4, 5, 6, 8 and 9 if no repetition is allowed?

The following table shows the income distribution of 600 families. Find the minimum income of the riches 30% families. Also the limits of income of middle 50% of families, to the nearest rupees. Income Below 75 75- 1…

a) Suppose P (x, y) denotes the equation8, what will the truth values of the Propositions P (2,2), P (0,4).

Show that the relation R on Z × Z defined by (a, b) R (c, d) if and only if a + d = b + c is an equivalence relation. Note: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

Let 8Z be the set of all integers that are multiples of 8. Prove that 8Z has the same cardinality as 3Z, the set of all integers multiples of 3.

We assume the question is: What is mathematical foundation for computer science? Solution: The mathematical side concentrates on areas where computers are used, or which are relevant to computer science, namely algebra, general topo…

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(a) By the definition of piece arrow- P\downarrow Q=P↓Q= ~(P\lor Q)(P∨Q) P\downarrow QP↓Q =~(P\lor P)(P∨P) We have derived that P\downarrow PP↓P is logically equivalent with ~P ~P=P&b…

Show that the following logical equivalences hold for the Peirce arrow ↓, where P ↓ Q ≡ ∼(P ∨ Q). a. ∼P ≡ P ↓ P b. P ∨ Q ≡ (P ↓ Q) ↓ (P ↓ Q) c. P ∧ Q ≡ (P ↓ P) ↓ (Q ↓ Q) H d. Write P → Q using Peirce arrows on…

F. Show using the rules of resolution/inference, that no single assignment of truth values to p, q, r makes all the disjunctions p V-9, p V-9,9 V r, qVT, V revaluate to true. Proof using truth na

F. How many different sequences, each of length r, can be formed using elements from A if (a) elements in the sequence may be repeated? (b) all elements in the sequence must be distinct?

Given the following 2 premises, 1. 𝑝 → (𝑞 ∨ 𝑟) 2. 𝑞 → 𝑠 Prove 𝑝 → (𝑟 ∨ 𝑠) is valid using the Proof by Contradiction method.

Use the Principle of Mathematical Induction to prove that (2𝑛)! /(2n)𝑛! is odd for all positive integers.

(P^Q^R)V (¬P^Q^R)V(¬P^¬Q^¬R)

Use the truth tables method to determine whether (¬p ∨ q) ∧ (q → ¬r ∧ ¬p) ∧ (p ∧ r) is satisfiable.

Suppose that A,B and C are sets such that A is the improper subset of B and B is the improper subset of C

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Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2 also symmetric?

Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive. b. irreflexive, symmetric, and transitive. c. irreflexive, antisymmetric, and not transitive. d. reflexive, neither symmetric nor antisy…

Justify whether the given operations on relevant sets are binary operations or not. - Multiplication and Division on set of natural numbers - Subtraction and Addition on Set of natural numbers - Exponential operation:…

Use the Principle of Mathematical Induction to prove that ((2n) !)/(2^n n !)((2n)!)/(2 n n!) is odd for all positive integers.

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1. Write the multisets (bags) of prime factors of given numbers. i. 160 ii. 120 iii. 250 2. Write the multiplicities of each element of multisets (bags) in Part 2-1(i,ii,iii) separately. 3. Determine the…

4. Let P(n) be the statement that 1+2+...+n'3D (n(n + 1)/2)2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do…

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Given the following 2 premises, 1. 𝑝→(𝑞∨𝑟) 2. 𝑞→𝑠 Prove 𝑝→(𝑟∨𝑠) is valid using the Proof by Contradiction method.

Draw the Hasse diagrams of all partial ordered sets with at most 4 elements. Which of these are lattices?

Determine whether each of these functions is a bijection from R to R (real).a) f (x) = 2x + 1b) f (x) = x2 + 1c) f (x) = x3

Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2 also symmetric?

show that 12+32+52+.....(2n+1)2=(n+1)(2n+1)(2n+3)/3 where n is a nonnegative integer.

A Function f : Z→Z, f(x) = x+5, is invertible since it has the inverse function g : Z→Z, g(x) = x−5.

Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cr…

An electrician charges a base fee of $70 plus $50 for each hour of work. Create a table that shows the amount the electrician charges for 1,2,3, and 4 hours of work. Let x represent the number of hours and y represent the amount charg…

Given The Following 2 Premises, 1. 𝑝→(𝑞∨𝑟) 2. 𝑞→𝑠 Prove 𝑝→(𝑟∨𝑠) Is Valid Using The Proof By Contradiction Method. [Hint: Use A Combination Of Equivalence Laws And Rules Of Inference To Solve This Question]

Construct the network who's activity and the relationship Activity:A,D,E can start simultaneously Activity:B,C>A;G,F>D,C;H>E,F

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Construct the network who's activity and the relationship Activity:A,C,D can start simultaneously Activity:E>B,C;F,G>D;H,I>E,F;J>I,G;K>H;B>A

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