Solved: Write the following statement in symbolic form using quantifiers: I. All students have taken a course in Mathematics II. Some students are intelligent, but not hardworking.

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]

Find the domain and range of this functions a) The function that assigns to each pair of positive integer the maximum of these two integers b) The function that assigns to each positive integer the number of the digits 0,1,2,3,4,5,6…

which of the following statements are true? justify your answers i) the contrapositive of "not a ⇒ not b' is "a&b', where a and b are two statements. ii) any set can be represented by the listing method

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 …

is 6x+1 < (2x+4) + (4x-2) proposition?

For all integers a, b and c, if 𝑎|𝑏 and 𝑏|𝑐, then prove that 𝑎𝑏 2 |𝑐 3 .

What is the image (range) of the function that assigns the square of an integer to this integer

Construct the call graph for a set of seven telephone numbers 555-0011, 555-1221, 555-1333, 555-8888, 555-2222, 555-0091, and 555-1200 if there were three calls from 555-0011 to 555-8888 and two calls from 555-8888 to 555-00…

prove a --> ( b V c ) using contradiction method and combination of inference rules and equivalence laws from these premises : 1. a --> ( d V b ) 2. d --> c

Use rules of inference to show that the hypothesis "All lions are fierce","Some Lions do not drink coffee" imply the conclusion "Some fierce creatures do not drink coffee".

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…

Assume that the function y(x) = f (x) + g(x) has a single local minimum in the interval 0 <= x <= 1, at x = xm. Write (i) a function M-file (called funct.m) and (ii) a MATLAB statement (using the fminbnd command) that together wil…

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)

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-w…

Each of 39 of my friends has either a dog a cat or a rabbit each of 24 of them has a dog each of 17 gas a cat and each of 16 has a rabbit the number having both a dog and a cat is one more than the number having both a cat and a rabbi…

An online shopping platform has given two types of saving to its members who have completed 2 doses of vaccination for Covid-19 when they purchase the products online, which are either discount of 15% or cashback of 15%. For the selec…

an + an-1 - 10an-2 + 8an-3

{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?

{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} 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 arrow…

Solution: injective function Definition: A function f: A → B is said to be a one - one function or injective mapping if different elements of A have different f images in B. A function f is injective if and only if whenever f(x) = f…

{F} 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

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

{F} 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 an…

Determine wheter 4 is the element of each of the following sets A: P={1,2,3,4,5} B: R={prime numbers}

Given P(x,y) :y=x+5 ,determine the truth value of each of the following if the universe is the set of real numbers.

Identify the mapping diagram that represents the relation and determine whether the relation is a function(−3,−6),(−1,−6),(5,−6),(8,−6)

A survey of 100 students produced the following information: 30 speak English, 45 speak Hindi and 60 speak Marathi. It is found that 10 speak both English and Hindi, 20 speak both Hindi and Marathi, 15 speak both English and Marathi a…

4. A survey of 100 students produced the following information: 30 speak English, 45 speak Hindi and 60 speak Marathi. It is found that 10 speak both. English and Hindi, 20 speak both Hindi and Marathi, 15 speak both English and Marat…

In a survey of 130 people, the following data were collected: 106 people subscribed to the newspaper, 29 people subscribed to magazines, and 17 people were members of a mail CD club. Seventeen subscribed to both the newspa…

How many distinct permutations are there in each word? 20. MOMMY 21. REARRANGE 22. MATHEMATICS 23. PROBABILITY 24. STATISTICS 25. PHILIPPINES

1. There are 41 candidates approved to be allotted as the F.D.R.E. council of cabinets. Historic facts of the candidates are shown below, 8 were served as former cabinet members,14 were university presidents, 15 served as members …

Solve the following set of recurrence relations and initial conditions3an-1+n^2-3,n>1,a0=1

What is mean by logic

Let R br a realtion defined from the set A={1,2,3,4} to the set B={2,3,4,5}as a€A, b€B aRb <-> a+b=5 1. What are the orderd pairs in the relation R 2. Represent R with a matrix

There are three kinds of people on an island: knights who always tell the truth, knaves who always lie, and spies who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight,…

an=3an−1+n2−3,n≥,a0=1

Show that if any five numbers from 1 to 8 are chosen, then two of them will add up to 9.

if R={(1,2),(2,1),(3,1),(2,3)} be a relation defined on A={1,2,3)then transitive closure of R is

Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find A ∩ B ∩ C. A ∪ B ∪ C. (A ∪ B) ∩ C. (A ∩ B) ∪ C.

Show that if A, B, and C are sets, then A ∩ B ∩ C = A ∪ B ∪ C by showing each side is a subset of the other side. using a membership table.

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

If A={1,2,3} , find the relation R={(a, b) ∈ A²|a>b}

Suppose that the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Express each of these sets with bit strings where the ith bit in the string is 1 if i is in the set and 0 otherwise. {3, 4, 5}=0011100000 {1, 3, 6, 10} = 101…

Venn diagram (A ∩ B) ∪ (A ∩ C) (A ∩ ) ∪ (A ∩ )

Show that if A, B, and C are sets, then A ∩ B ∩ C = A ∪ B ∪ C •by showing each side is a subset of the other side. •using a membership table.

Let p and q be propositions, construct the truth table for the compound proposition. ~(p^q)

What is the variable x after the statement -if2+3=6 if and only if 3+2=5, x = x+1^ - if x = 2 ?

Let A represent the set of all students at a university, and let B represent the set of all courses offered at the university. What is the Cartesian product A × B and how can it be used?

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

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…

prove a --> ( b V c ) using contradiction method and combination of inference rules and equivalence laws from these premises : 1. a --> ( d V b ) 2. d --> c

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

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

State the converse and contrapositive of each of the following implications. (a) If it does not rain tonight, I will go fishing tomorrow. (b) If it rains tonight, then I will stay at home.

Let A={0, 1,2}×{2, 5,8}={(0, 2), (0, 5), (0, 8), (1, 2), (1, 5), (1, 8), (2, 2), (2, 5), (2, 8) }. A partial order relation R on A is defined by (a, b) R(c, d) if and only if (a+b) divides (c+d). Draw a hasse diagram for poset A.

Let A = {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15} and define relations R and S from X to Y as follows:  for all (x,y) ∈A × B, x R y ⇔ y % x  for all (x,y) ∈A × B, x S y ⇔ (y – 4)+2 = x State explicitly which ordered pai…

The market for lemon has 10 potential consumers, each having an individual demand curve P = 101 - 10Qi , where P is price in dollars per cup and Qi is the number of cups demanded per week by the i th consumer. Find the mar…

Let A = {1,2,3,4,5,6} and let p1 = (3,6,2) and p2 = (5, 1, 4) be permutations of A. (a) Compute p1 ◦ p2 and write the result as a product of cycles and as the product of transpositions. (b) Compute p−1 ◦ p−1

Let A = {a,b,c,d}, B = {1,2,3}, and R = {(a,2), (b, 1), (c, 2), (d, 1)}. (a)Is R a function? (b)Is R−1 a function? Explain your answers. Expert's answer

Draw hasse diagram representing the positive divisor of 36 and show digraph to It?

Question 2: Consider the following narration: If equality of opportunity is to be achieved then those people previously disadvantaged should be given special opportunities. If those people previously disadvantaged should be given s…

Suppose that, in the circuit for a 2-bit comparator, we wanted to allow for inputs from the higher byte comparisons, xH < yH, xH = yH, xH > yH, so that we could now use a 2-bit comparator in cascades for building comparators for num…

a) Consider the full adder function: sum(c i-1, xi , yi ) = (ci , si ) Where xi , yi are the ith bits of the binary numbers X and Y, ci-1 is the carry in and the outputs are: ci , the carry out and si is the sum output. Write…

question 2 Consider the truth function f(P, Q, R) = (P \rightarrow Q) \rightarrow R a) Find a restricted statement form in conjunctive normal form (CNF) logically equivalent to f. b) Find a restricted statement form i…

a) Consider the whole of the English words set. Suppose an English word x is related to another English word y if x and y begin with the same letter. i) Show that this is an equivalence relation. ii) Compute C(quadratic) and C(rhom…

a) Consider the following functional relation, f, defined as: f : R \rightarrow R, f(x) = x2 Determine whether or not f is a bijection. If it is, prove it. If it is not, show why it is not. b) Consider the set F = {…

a) Consider the whole of the English words set. Suppose an English word x is related to another English word y if x and y begin with the same letter. i) Show that this is an equivalence relation. ii) Compute C(quadratic) and C(rhombus…

a) Consider the full adder function: i-1 i i i i sum(c , x , y ) = (c , s) Where xi i i-1 , y are the i bits of the binary numbers X and Y, c is the carry in and the outputs are: th ci i , the carry out and s is the sum output. Write …

Prove or disprove the validity of the argument"every living things is a plant or an animal",Kamal's dog is alive and it is not a plant",All animals have heart",Hence "Kamal's dog has a heart".

Let A = {a,b,c,d}, B = {1,2,3}, and R = {(a,2), (b, 1), (c, 2), (d, 1)}. (a)Is R a function? (b)Is R−1 a function? Explain your answers.

A group of students study at least one of the following subjects:biology, chemistry and physics. 36 of them study chemistry, 39 study biology and 41 study physics.15 study physics. 9 study all the three subjects. @draw a Venn diagram …

Write the statements in symbolic form , and  and the indicated letters to represent compound statements. a) Let s = “stocks are increasing” and i = ”interest rates are steady”. i.…

Let A = {x € N: 3 ≤ x ≤ 13}, B = {x € N : x is even}, and C = {x EN: x is odd}. (a) Find An B. (b) Find A UB. (c) Find Bn C.. (d) Find BUC.

Draw a Venn diagram to represent each of the following: (a) AUB (b) (AUB) (c) An (BUC) (d) (ANB) UC (e) AnBnC

Show that (p → q) ∧ (p → r) and p → (q ∧ r) are logically equivalent using logical equivalence laws.

How many ways are there to select 15 countries in the United Nations to serve on a council if 4 are selected from a block of 35, 5 are selected from a block of 40, and the others are selected from the remaining 55 countries?

Symbolic Logic Compute the truth values of the following, given that A, B, and C are true, and X, Y, and Z are false. 1. ~ [((A⊃J ~B) ⊃J~C) ⊃J~X] 2. [(((A⊃J B) ⊃J X) ⊃J Z) ⊃J Y] ⊃J [((X ⊃J Z) ⊃J A) ⊃J X] Compute the truth va…

Symbolic Logic Can you figure out the values of the following below? Why or why not 1. (H ≡≡ A) ⊃J (G ≡≡ B) 2. ~ (X· G) For each of the following pairs of sentences below, indicate whether (1) is a sufficient condi[1]tion fo…

Write the following into symbolic form: Some student in this class has studied programming.

In how many different ways can 3 elements be selected in order from a set with five 10 when repetition is allowed?

Convert the following sentences into symbolic form. i) All planets revolves around sun in an elliptical orbit ii) No student who is intelligent will fail.

What is the probability of these events whenwe randomly select a permutation of the 26 lowercase letters of the English alphabet? a) The first 13 letters of the permutation are in alphabetical order. b) a is the first letter of the …

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

Prove that 1·1 ! + 2·2!+· ··+ n · n! = (n+I)!-1 whenever n is a positive integer.

Show that (¬𝑃 ⟶ 𝑅) ∧ (𝑄 ↔ 𝑃) ⟺ (𝑃 ∨ 𝑄 ∨ 𝑅) ∧ (𝑃 ∨ ¬𝑄 ∨ 𝑅) ∧ (𝑃 ∨ ¬𝑄 ∨ ¬𝑅) ∧ (¬𝑃 ∨ 𝑄 ∨ 𝑅) ∧ (¬𝑃 ∨ 𝑄 ∨ ¬𝑅).

|4| >2 if -2 <1<2

Let P(x) be the statement “x spends more than five hours every weekday in class,” where the domain for x consists of all students. Express each of these quantifications in English. a) ∃𝑥𝑃(𝑥) b) ∀𝑥𝑃(𝑥) c) ∃𝑥…

List all the ordered Paris in the relation R=[(a, b) |a divides b ] on the set {1,2,3,4,5,6}.

1) Translate the given statement into propositional logic using the propositions provided You cannot edit a protected Wikipedia entry unless you are an administrator. Express your answer in terms of e: “You can edit a protected Wi…

Write the following statement in symbolic form using quantifiers: I. All students have taken a course in Mathematics II. Some students are intelligent, but not hardworking.

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