Solved: Question 1. [20 Marks] a) Let p, q, and r be the propositions: p = "the flag is set" q = "I = 0" r = "subroutine S is completed" Translate each of the following propositions into symbols, using the letters p, q, r and logical connectives. (i) If the flag is set, then I = 0 [2 marks] (ii) The flag is set and I = 0 if subroutine S is not completed [2 marks] (iii) Subroutine S is completed if and only if I = 0 and flag is set [2 marks] b) State the converse, contrapositive, and inverse of each of these conditional statements. (i) If it snows tonight, then I will stay at home [3 marks] (ii) I go to the beach whenever it is a sunny summer day [3 marks] (iii) When I stay up late, it is necessary that I sleep until noon [3 marks] c) Explain the step-by-step procedure involved in finding the inverse of an n by n square matrix. [5 marks]

a) Draw a tree with n vertices with n+1 vertices of degree 2, n+2 vertices of degree 3, and n+3 vertices of degree 1. Where n is even digit of your arid number e.g 19-arid-234 take n=2

(b) For any natural number n, prove the validity of given series by mathematical induction:

In a school 844 students have access to three software packages A, B, C. Where 743 didn’t use any software, 740 used only package C, 742 used only package A,741 used package B, 739 used all three packages, 738 used both A and C …

Produce a truth table for given Boolean expression (A+B'+C)(A+B+C)(A'+B+C')

Let R and S be relations on X. Determine whether each statement is true or false. If the statement is true, prove it; otherwise, give a counterexample. 1. If R is transitive, then R−1 is transitive. 2. If R and S are refl…

1. Three departmental committees have 6, 12, and 9 members with no overlapping membership. In how many ways can these committees send one member to meet with the president? 2. Two dice are rolled, one blue and one red. How ma…

Basic Counting Principle 6. How many different car license plates can be constructed if the licenses contain three letters followed by two digits if: a.) Repetitions are allowed; b.) repetitions are not allowed. 7. Two d…

Find the generating function of recurrence relation an+1_an=3n ,n less than 0 where ao=1

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?

Show that ¬ (P\iff⟺Q)\iff⟺(P V Q) Λ ¬(P Λ Q) \iff⟺(P Λ ¬Q) V (¬ P Λ Q) without using truth table

Design a single error correcting code for m=3 & n=7

(a) In how many ways can a committee of 3 faculty members and two students be selected from 7 faculty members and 8 students (b) How many ways are there to distribute 12 different books among 15 people if no person is to receive more…

(a) Show that a simple connected graph with 7 vertices each of degree 4 is non-planar (b) Find χ(Kn) & χ(Cn)

Let g be a function from Z+ (the set of positive integers) to Q (the set of rational numbers) defined by (x, y) element of g iff y = (g is a subset of Z+ mapped with Q) and let f be a function on Z + defined by (x, y) element of f iff…

Let A = {1,2,3,4,6,8,9,12,18,24} be a non-empty set and R be the partial order relation of divisibility defined on A, i.e., If (a,b) ε A, then a divides b i. Draw the Hasse diagram of R. ii. Find the Maximal and Minimal elements in…

Use breadth-first search to produce a spanning tree for each of the simple graphs in Exercises 1 3-15. Choose a as the root of each spanning tree.

an-3an-1-4an-2=4.3n

Solve the Recurrence relation an-3an-1-4an-2=4.3n where a0=1,a1=2

Solve the Recurrence relation an-3an-1-4an-2=4*3n where a0=1,a1=2

Out of 300 students taking discrete mathematics, 60 take coffee, 27 take 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…

Let W ={1,2,….,8} Q = {2,4,6,8,10}, Y = {1,2,4,5,6,8,9}. Evaluate: W union Y (2 marks) Q intersection Y (2 marks) Set Difference P minus Y (2 marks) (W intersection Q) union Y

Let p, q, and r be the propositions: p = "the flag is set" q = "I = 0" r = "subroutine S is completed" Translate each of the following propositions into symbols, using the letters p, q, r and logical conn…

Develop a digital circuit diagram that produces the output for the following logical expression when the input bits are A, B and C i. (A ∧ B) ∨ ((B ∧ C) ∧ (B ∨ C)) [4 marks] ii. (A ∧ B ∧ C) ∨ A ∧ (￢ B ∨ ￢C) [4 mark

Let U = {l, 2, 3, 4, 5, 6, 7, 8, 9, and 10} be a universal set. Let A, B, C such that A= {l, 3, 4, 8}, B = {2, 3, 4, 5, 9, 10}, and C = {3, 5, 7, 9, 10}. Use bit representations(computer representation) for A, B, and C together with …

A number of students prepared for an examination in physics, chemistry and mathematics. Out of this number, 15 took physics, 20 took chemistryand 23 took mathematics, 9 students took both chemistry and mathematics, 6 student took both…

Determine truth value for this statement if the domain consist of all integers Vn( n+1>n)

Q1. Explain complement arithmetic and its significance in the computation. Q2. Compute the value of AFD416-BECE16+67758 using signed magnitude representation Q3. Evaluate the expression in Q2. above using 1's complement arithmetic…

(a) Suppose f and g are functions whose domains are subsets of z+, the set of positive integers. Give the definition of 'f is \OmicronO(g)' (b) Use the definition of 'f is \OmicronO(g)' to show that: (I) 2n+27 is \O…

If a simple graph G has p vertices and any two distinct vertices u and v of G have the property that degGu + degGv ≥ p-1 then prove that G is connected

Solve recurring relation :an+2 -10an+1+25an=5n(n≥0)

Show that whether x5 + 10x3 + x + 1 is O(x4) or not?

Write smallest cutset of K3,5

Let f: A\to→ B and g: B\to→ C be functions. Show that if g o f is onto, then g is onto.

Let f:R\to→ R be f(x)= x/1+|x| a) Is f everywhere defined? If not, give the domain. b) Is f onto? If not give the range. c) Is f one-to-one? Explain. d) Is f invertible? If so, what is f-1?

Consider the following relations on the set A={1,2,3,4,5}. a) R1={(1,1), (1,2), (3,3), (3,2)} b) R2={(1,1), (2,2), (3,2), (4,5), (5,2)} c) R3={(1,4), (2,5), (3,3), (4,2), (5,2)} State, with reasons, for a), b), c) whether the rela…

Question #221673 Say for each of the posets represented by the given Hasse-diagram whether the poset is i) a lattice ii) a complemented lattice iii) a Boolean algebra Give reasons for your answers

Describe Konigsberg Bridge problem

Branches and chords may change according to the spanning tree.”-- Do you agree? Justify your answer

Given the following recurrence relation (M). an = −4an−1 + 5an−2, a0 = 2, a1 = 8 The solution of (M) is: a. an = 3 − (−5) n b. an = 3 + (5) n c. an = (3) n − 5 d. None of these

ACTIVITIES/ASSESSMENT: A. Tell if the following statements are propositions or not. 1. Study hard! 2. The Apple Macintosh is a 16-bit computer. 3. 1 is an even number. 4. Why are we here? 5.8+7=13 B. p is "x …

Question #222539 1. Let p and q be the propositions “Swimming at the New Jersey shore is allowed” and “Sharks have been spotted near the shore,” respectively. Express each of these compound propositions as an English sentence. a) ¬…

let z be the set of integers and R be the relation on Z defined as: aRb if and only if 1+ab>0 then

What is degree of a vertex in a graph?

Express the negations of each of these statements so that all negation symbols immediately precede predicates. ∃x∃y(Q(x, y) ↔ Q(y, x))

Draw the Venn diagrams for each of these combinations of the sets A, B, C, and D. *A̅ ∪ B̅ ∪ C̅ ∪ D̅

Express each of these statment using quantifires : a) every student in this classes has taken exactly two mathematics classes at this school. b) someone has visited every country in the world except Libya

Describe the Hasse diagram formed by the Relation "x is a divisor of y" for the set A = {1, 3, 6, 12, 24, 48}

Express each of these statment using quantifires : a) every student in this classes has taken exactly two mathematics classes at this school. b) someone has visited every country in the world except Libya

Express each of these statement using quantifires : No one has climbed every mountain in the Himalayas

Express each of these statement using quantifires : every movie actor has either been in a movie with Kevin Bacon or has been in a movie with someone who has been in. a movie with Kevin Bacon.

3.3 In a class of 100 learners, 80 do Mathematics and 35 do Physics. If x learners do Mathematics and Physics, and y learners do neither Mathematics nor Physics, find the greatest possible value of y.

In the following Statement: " If the BIOS test runs fine, the CPU and motherboard must be OK. If the CPU and motherboard and memory are all OK, then there must be a flaw in the OS. The BIOS test runs fine and the memory is OK. …

Express 2x+1 / x(x+1) in partial fractions and hence find the general solution of the differential equation x(x+1)dy/dx = y(2x+1) expressing y explicitly in terms of x.

A Hasse diagram is a graphical rendering of a partially ordered set displayed via the cover relation of the partially ordered set with an implied upward orientation. A point is drawn for each element of the poset, and line segments ar…

Describe the Hasse diagram formed by the Relation "x is a divisor of y" for the set A = {1, 3, 6, 12, 24, 48}

For the following relation on the set {1,2,3,4,5,6,7,8,9,10,11,12}.

Describe the Hasse diagram formed by the Relation "x is a divisor of y" for the set A = {1, 3, 6, 12, 24, 48}

Find the number of distinct permutation of the word 'programmer '

Draw the Venn diagrams for each of these combinations of the sets A, B, and C. a) A ∩ (B − C) b) (A ∩ B) ∪ (A ∩ C) c) (A ∩ B) ∪ (A ∩ C)

The number of elements in the Power set P(S) of the set S = { { Φ} , 1, { 2, 3 }} is?

complete \space question \space is \space \\ Let \space f:R→R \space is \space defined \space by \space f(x)= \space \frac{x}{1+∣x∣}.\\ \space Then \space…

Let f :R \to→ R be f (x) =x /(1+|x|)

2) Translate the given statements into propositional logic using the propositions provided You are eligible to be the President of the USA if and only if you are at least 35 years old, were born in the USA, or at the time of your …

Suppose the city of MISSISSAUGA has a contest where they arrange the letters of their city name in a particular order. The city residents have to guess which order the city has chosen, and the winner gets $10 000. The rules allow peop…

Determine whether the function f(x) = x^2 + 3x + 2 from Z to Z is a bijective function.

Solve the recurrence by master's method. a) T(n) = 3T[n/4)+ cn^2 b) T(n)=T[2n/3)+1

Solve the recurrence by substitution method. T(n) = 2T (n/2)+n-1 and T (1) =1.

Translate the following English sentences into Propositional Logic. Let: C A bird can fiy F A bird has wings a) If Bird has wings, then bird can fly. b) If bird has no wings, then it can't fly.

Translate the following Propositional Logic to English sentences. Let: E-Lion is eating H=Lion is hungry a) E-H b) EA-H c)-(H-E)

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Solve the recurrence by substitution method. T(n) = 2T (n/2)+n-1 and T (1) =1.

Draw the Venn diagrams for each of these combinations of the sets A, B, C, and D. a) (A ∩ B) ∪ (C ∩ D) b) A c U B c U Cc U Dc c) A − (B ∩ C ∩ D) d) A ⊕ B

How many strings of six lowercase letters from the English alphabet contain: i.The letter a? ii. The letters a and b? iii. The letters a and b in consecutive positions with a preceding b, with all the letters distinct? iv. The letters…

Let A= {0, 1, 2, 3} and define relations R, S and T on A as follows: R= { (0,0),(0,1),(0,3),(1,0),(1,1),(2,2),(3,0),(3,3)} S= {(0, 0),(2,2),(1,1), (0, 2), (0, 3), (2, 3),(2,2)} T= {(0, 1), (2, 3),(0,0),(2,2),(1,0),(3,3),(3,2)} i. Is R…

Determine whether the relation R on the set of all people is irreflexive, where (a, b) ∈ R if and only if : i) a and b were born on the same day. ii) a has the same first name as b. iii) a and b have a common grandparent

Question No 05: [07 marks] Let f (x) = ax + b and g(x) = cx + d, where a, b, c, and d are constants. Determine necessary and sufficient conditions on the constants a, b, c, and d so that f ◦ g = g ◦ f

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Prove that for every positive integer n. 1.2.3+2.3.4++ n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4.

How many strings of six lowercase letters from the English alphabet contain: i.The letter a? ii. The letters a and b? iii. The letters a and b in consecutive positions with a preceding b, with all the letters distinct? iv. The lett…

Determine whether the function f(x) = |4x| from Z to Z is a bijective function.

A boy has 10 red balls, 20 blue balls,25 black balls ,and 30 pink balls .He select ball at random without looking at them .calculate the minimum number of balls he must select to be sure that at least 6 balls of the same color .

1) Draw the Hasse diagram for inclusion on the set P(S), where S = {a, b, c, d} 2) Let S = {1,2,3,4} with lexicographic order "<=" relation a. Find all pairs in S x S less than (2, 3) b. Find all pairs in S x S greater than…

Determine how many bit strings of length 5 can be formed ,where three consecutive 0s are not allowed

Describe the Hasse diagram formed by the Relation "x is a divisor of y" for the set A = {1, 3, 6, 12, 24, 48}

Determine whether the given relation is reflexive, symmetric, transitive, or none of these. R is the ”greater than or equal to” relation on the set of real numbers: For all x, y ∈ R , xRy ⇐⇒ x ≥ y

determine how many bit strings of length can be formed, where three consecutive 0s are not allowed

Use a Venn diagram to illustrate the set of all months of the year whose names do not contain the letter R in the set of all months of the year.

Let p and q be the propositions defined as below. p : It is below freezing. q : It is snowing.

Show that a complete graph with n vertices has n(n -1) 2 edges

Show that for any real number x, if x2 is odd, then x is odd.

Let p, q and r be statements. Use the Laws of Logical Equivalence and the equivalence of → to a disjunction to show that: ∼ ((p ∨ (q →∼ r)) ∧ (r → (p∨ ∼ q))) ≡ (∼ p ∧ q) ∧ r.

Let p, q and r be statements. Suppose you know that the statement form ((q → p) ∨ r) ∨ (∼ r ∧ p) is false. What can you conclude about the truth values of the three statement variables?

Model two contextualized problems using binary trees both quantitatively and qualitatively

Prove the following sentences using any prove method [ the most appropriate one]: a) If r is rational and s is irrational, prove that 2r+s is irrational. b) If t and s are integers and t × s is even, then t is even or s is even.

Prove that (a ∧ (b → ¬a)) → ¬b is a tautology.

Prove that the following argument is valid. Write all the necessary steps a)Rashid didn’t perform well in the subject, but he was present in every class. b)Every student who completed all the assignments, performed well in the subje…

Determine whether each of the following compound propositions is satisfiable. Justify your answer. 1. (a) (pv-g)^(-pVg)^(-p^ g) (b) ()^(-)^(-p)^(---) (c) (pvqv-r) ^ (pv-gv-8) ^ (pv-rv-8)^(-pv-gv-) A (pvqV-s)

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