Solved: (a) How many bit strings of length 8 are there? (b) How many bit strings of length 8 or less are there? (Count the empty string of length zero also.) (c) How many strings of 6 lower case English letters are there that have the letter x in them somewhere? Here strings may use the same letter more than once. (Hint: It might be easier to first count the strings that don't have an x in them.) (d) A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in 2 sizes for each sex. How many different types of this shirt are made?

Suppose a recurrence relation an=7an−1−12an−2 where a1=16 and a2=52 can be represented in explicit formula, either as: Formula 1: an=pxn+qnxn or Formula 2: an=pxn+qyn where x and y ar…

Show that the relation R = ∅ on a nonempty set S is sym- metric and transitive, but not reflexive.

Show that the relation R = ∅ on the empty set S = ∅ is reflexive, symmetric, and transitive.

Represent each of these relations on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order). a) {(1, 1), (1, 2), (1, 3)} b) {(1, 2), (2, 1), (2, 2), (3, 3)} c) {(1, 1), (1, 2), (1, 3), (2, 2), (…

Draw the directed graphs representing each of the relations on {1, 2, 3, 4} a) {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} b) {(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)} c) {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (…

Answer these questions for the poset ({3, 5, 9, 15, 24, 45}, |). a) Find the maximal elements. b) Find the minimal elements. c) Is there a greatest element? d) Is there a least element? e) …

What are the values of the following expressions? ∑_(i=3)^6▒ (i^2+6) ∏_(j=1)^5▒ (j/2) 4! 0! (6¦2) C(6,2)

Show that if n is a positive integer with n ≥ 3, then C(n, n − 2) = ((3n − 1)C(n, 3))/4

A gumball machine contains 300 grape flavored balls, 400 cherry flavored balls, and 500 lemon flavored balls. What is the probability of getting 1 grape ball, 1 cherry ball, and 1 lemon ball if each ball was removed and then replaced …

Let A = {2, 4} and B = {2, 4, 6} and define a relation R for A to B as follows: Question: Which ordered pairs belongs to relation R?

Construct a K-map for F(x, y, z) = xz + yz + xyz. Use this K-map to find the implicants, prime implicants, and essential prime implicants of F(x, y, z).

1. Classify and compare the different types of antenna using ah venn diagram. Discuss each type in each petal of the diagram.

1a) Use set-builder notation to prove that A-(B ∩C)=(A-B)∪(A-C) b) Let E={ 1,23,4,5,6,7,8,9,10} and ordering of element in increasing order that is a= i; what bit strings represent the subsets of integers not exceeding 5 in E …

Describe any three inference rules

Write short note on Caesar cipher with example

Show that every simple finite graph has two vertices of the same degree

Define spanning trees in a weighted graph, with an example

Show that a tree with n vertices has exactly n-1 edges

Let the operations * and ⊕ be defined on the set of integers. Find each of the following are commutives, associatives, left-distributive, and right-distributive. 1. a*b=ab; a⊕b=-a-b 2. a*b=a+2b; a⊕b=a-2b 3. a*b=2ab; a⊕b=2a+2…

Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation defined by aOb = max{ a,b}

Use generating functions to solve the recurrence relation an = 4an−1 − 4an−2 +n2 , where a0 = 2, a1 = 5.

Write each of these statements in the form “if p, then q” in English. a) It snows whenever the wind blows from the northeast. b) The apple trees will bloom if it stays warm for a week. c) That the Pistons win the championship im…

A house which is valued at 2,600, 000 appreciated at rate of 12% per year I. What will be its value after two and half years? ii. After how long will its value be 4200,000

Consider the following series 56, 28, 14.. I. Find 17th term. ii. Find the sum of the series if it continues indefinitely iii. Find 20th term.

Show that (p → r) ∧ (q → r) and (p ∨ q) → r are logically equivalent.

Show that each of these conditional statements is a tautology by using truth tables. a) (p ∧ q) → p b) p → (p ∨ q) c) ￢p → (p → q) d) (p ∧ q) → (p → q) e) ￢(p → q) → p …

What is the truth value of ∀xP(x), where P(x) is the statement “x2 < 10” and the domain consists of the positive integers not exceeding 4?

Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.] a) It is necessary to wash the boss’s car to get promoted. …

Suppose you want to encode messages containing only the following characters with their given respective frequencies: B: 55 D: 15 E: 80 G: 5 U: 45 (a) What is the minimum length bit string required to encode each character with a dis…

Let us take a connected multi graph G with 2k vertices of odd degree and go in reverse direction for the proof. Initially we start pairing the vertices of odd degree and go on adding an extra edge joining the vertices in each pair, i…

Suppose that G is a connected multigraph with 2k vertices of odd degree. Show that there exist k subgraphs that have G as their union, where each of these subgraphs has a Euler path and where no two of these subgraphs have an edge in …

Consider the nonhomogeneous linear recurrence relation an = 3an−1 + 2^n. Show that a^n = – 2^(n+1) is a solution of this recurrence relation.

Determine values of the constants A and B such that an = An + B is a solution of recurrence relation an = 2an−1 + n + 5. Hence, find the solution of this recurrence relation with a0 = 4.

Pick a number. Add 4 to the numbers, and multiply the sum by 2 subtract 5 and decrease this difference by twice the original numbers (hint let n represent the original numbers and 2n be twice

List five situation from daily life in which graphs arise naturally

j=0 ∑ 8 ( j 8 ) (j+1)(j+2) 1 =1⋅ (0+1)(0+2) 1 +8\cdot\dfrac{1}{(1+1)(1+2)}+28\cdot\dfrac{1}{(2+1)(2+2)}+8⋅ (1+1)(1+2) 1 +28⋅ (2+1)(2+2) 1 +56\cdot\dfrac{1}{(3+1)(3+2)}+70\cdo…

∑j=08(j8)(j+1)(j+2)

Consider series 56,28, 14.. I. Find 17th term Ii. Find the sum of series if it continues indefinitely. Iii.find 20th term Ii. Find the sum of series if it continues indefinitely. Iii.find 20th term

List the quadruples in the relation { a,b,c,d} where a, b, c, d are integers with 0 < a < b < c < d < 8

Let f : A → B be a function. 1. Show that for the identity function iA on A we have f ◦ iA = f. 2. Show that for the identity function iB on B we have iB ◦ f = f.

Show that a tree with n vertices has exactly n-1 edges

A. Try these! 1. Find a relation R such that 𝑥+𝑦 2 >1 if A = {0,1, 2} and B ={0, 1, 2, 3}. 2. Find a relation R such that y is a power of x if A = {1, 2, 3} and B = {1, 4, 5, 9}

1. Find a relation R such that 𝑥+𝑦 2 >1 if A = {0,1, 2} and B ={0, 1, 2, 3}.

Let f(x) = x – 3 , g(x) = 2x + 1 and h(x) = x2 – 5. Find the following 1. f ₒ g 2. f ₒ h 3. h ₒ g 4. g ₒ f 5. g ₒ h

B. Exercises: Let f(x) = x – 3 , g(x) = 2x + 1 and h(x) = x2 – 5. Find the following 1. f ₒ g 2. f ₒ h 3. h ₒ g 4. g ₒ f 5. g ₒ h

Determine the domain of each of the following functions: 1. f(x) = x + 10 6. A(x) = x2 -2 2. F(x) = 2 3 𝑥 + 5 7. H(x) = √𝑥 − 2 3. g(x) = 5 – 3x 8. K(x) = √𝑥 2 − 2 4. g(x) = 1 (𝑥+5)(𝑥−1) 9. C(x) = 2x3 + 4x2 - 2x + 1 5. b(x) = 𝑥−1 𝑥 2+5…

Let A, B and C denotes the subset of a set, S and let 𝐶̅denotes the complement of C in set S. If (A ∩ C) = (B ∩ C) and (A ∩ 𝐶̅) = (B ∩ 𝐶̅), then prove that (A = B).

Given A = {x | x ∈ Z- ∧ 3x > -10}. Determine elements of A.

Let the universe of discourse be the set of all integers. Let p; q; r; s, and t be as follows: p(x):x>0,q(x):xiseven,r(x):xisaperfectsquare,s(x):xis(exactly)divisibleby4, t(x):x is (exactly) divisible by 5. (8 marks) Write the follow…

Construct a truth table for each of these compound propositions. (i) (p → q) ↔ (¬q → ¬p) (16 marks) ii) p ⊕ (p ∨ q) (8 marks) (i) Determine by using truth tables if (p ∧ q) → p is a tautology, contradiction or a contingency. Give…

How many ways are there to select 12 countries in the United Nations to serve on a council if 3 is selected from a block of 53, 1 are selected from a block of 62 and 8 are selected from the remaining 74 countries?

Showing all working, find a formula for the general term of the sequence (sn) = s3, s4, s5 . . . defined by s3 = π and sn = sn-1 - 2 for n >= 4.

Consider the sequence (wn) defined recursively by w2 = 3 wn = (wn-1 (1 - 0.02)) - 2(n + 1)/ n2 for n >= 3. Write a MATLAB program to compute wn for n = 2, 3, 4, . . . , 10 and display the values n and wn in two columns with app…

Let A be a given finite set and P(A) its power set. Let ⊆ be the inclusion relation on the elements of P(A). Draw Hasse diagrams of (P(A), ⊆) for A={a}; A={a,b}; A={a,b,c} and A={a,b.c.d}.

Which of the following is false about a proposition? (¬p ∨ p) is a proposition.(¬p ∧ p) is a proposition.A predicate is not a proposition.c.(¬p ∨ ¬p) is a proposition.

Translate the following statement into mathematical expression based on the given propositional variables. “If a country’s leaders are elected based on good leadership traits, then the country will have good governance”. p : A count…

Calculate the quartiles, (i,e., Q1 and Q3,), deciles D3 and D7, Percentiles P10, P23 and P92 for the following data. Also find the Quartile Range and Quartile Deviation. Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No…

The ages of 100 persons are tabulated below. Find Q1, Q3, D2, D8, P20 and P60. Above what age we have the oldest 10% persons? Age(in years) 10-20 20-30 30-40 40-50 50-60 60-70 70-80 No. of persons 16 20 21 28 10 3 2 Note: …

What is symbolic form of it is not true that, the students answered their activity or participated in the class discussion

what is symbolic form of she did nit received a promotion or she did not received a salary increase

3Fn − Fn−2 = Fn+2, for n ≥ 3.

2. A survey on subjects being taken by 250 college students in Candon City revealed the following information. Subject Number of students taking the subject Mathematics 145 Filipino 90 English 88 Math and Filipino 25 Filipin…

Produce truth tables for given Boolean expressions A’B’C + AB’C + ABC + ABC’

Consider the recurrence relation an= 3 an1 + 3 an-2 when ao = 1 and a1= 2 then a3= ..: O 56

A student can choose a computer project from one of three lists. The three lists contain 23, 15 and 19 possible projects, respectively. How many possible projects are there to choose from?

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

Let S = [1,2,4,5,10,20, 25, 50, 100). Then show that 5 forms a lattice under divisibility. Draw the Hasse diagram also.

Suppose we are given a Boolean function f(x, y, z)=(x+y)(x+y)(x+2). Find its DN form

Determine how many bit strings of length 6 can be formed, where the last bit is 0 and two consecutive 0s are not allowed.

Let A be a given finite set and P(A) its power set. Let ⊆ be the inclusion relation on the elements of P(A). Draw Hasse diagrams of (P(A), ⊆) for A={a}; A={a,b}; A={a,b,c} and A={a,b.c.d}.

Consider the sequence (wn) defined recursively by w2 = 3 wn = (wn-1 (1 - 0.02)) - 2(n + 1)/ n2 for n >= 3. Write a MATLAB program to compute wn for n = 2, 3, 4, . . . , 10 and display the values n and wn …

Find a relation R such that 𝑥+𝑦 2 >1 if A = {0,1, 2} and B ={0, 1, 2, 3}. 2. Find a relation R such that y is a power of x if A = {1, 2, 3} and B = {1, 4, 5, 9}

Let f : A → B, g : B → C, and h : C → D be functions. 1. State what you need to show to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f. 13 2. Consider now some a ∈ A. Calculate h((g ◦ f)(a)) and (h ◦ g)(f(a)). Are they equal? 3. U…

The chairs of an auditorium are to be labeled with a letter and a positive integer not exceeding 100. What is the largest number of chairs that can be labeled differently?

If 2 distinguishable dice are rolled, in how many ways can they fall? If 5 distinguishable dice are rolled, how many possible outcomes are there? How many if 100 distinguishable dice are tossed?

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?

How many numbers can be formed using the digits 1, 3, 4, 5, 6, 8 and 9 if no repetition is allowed? 13) Suppose that we draw a card from a deck of 52 cards and replace it before the next draw. In how many ways can 10 ca…

In how many ways can 10 people be seated in a row so that a certain pair of them are not next to each other? Three persons enter into car, where there are 5 seats. In how many ways can they take up their seats?

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

Exercise 9: Draw a full binary tree having the following properties 1. Four internal vertices and five terminal vertices. 2. Height = 3 and nine terminal vertices. 3. Height = 4 and nine terminal vertices.

Use any of the two proof methods to prove: ((~a^b)^(b^c))^~b

((~a^b)^(b^c)^~b

let q(x) denote the statement x is an integer . what are the truth values of the following?

If the statement q ^ r is true, determine all combinations of truth values for p and s such that the statement (q -> [¬p v s]) ^ [¬s -> r] is true.

a) Suppose P (x, y) denotes the equation y=x+8, what will the truth values of the Propositions P (2,2), P (0,4).

Ask about the lenght of 5 formed using the lettre ABCDEFG without repetitions. How many string begins with the letter F and end with the letter A

For each recurrence relation and initial conditions, find: (i) general solution; (ii) unique solution with the given initial conditions: (a) an = 3an−1 + 10an−2; a0 = 5, a1 = 11

If A= {1,3,4,6,7} B= {1,2,3,7,6,9,4} C={3,2,7,9,6,1,5,8,4} depict sets A,B and C in a Venn diagram

If x and y are not both even numbers, then neither the square of x and y is odd, nor the cube of x and y is even

Let R be a binary relation on N × N defined by (w, x ) R (y, z) if and only if w = y and x ≤ z . Is R reflexive? Is R symmetric? Is R antisymmetric? Is R transitive?

For each of the following sets, a) S={1,2,3}, T ={a, b, c} b) S={a, b}, T ={1,2,3,4} c) S={1,2,3,4}, T ={a, b} Determine whether 1. There is a one-to-one function f: S→T; 2. There is an onto function f: S→T; …

Proof that an undirected graph has an even number of vertices of odd degree.

Construct a K-map for F(x, y, z) = xz + yz + xyz. Use this K-map to find the implicants, prime implicants, and essential prime implicants of F(x, y, z).

From a total of 125 employees, 102 employees are assigned to perform ONLY ONE programming job, either system programming or application programming. And 10 employees are not assigned to perform any of those programming jobs. Overall, …

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