Solved: d) Assume ⊥ occurs in a subproof of the main proof. The appearance of ⊥ in this subproof indicates that: 1. The premises of the main proof are mutually inconsistent. 2. The assumption of the subproof is a TT-contradiction. 3. One of the premises is the negation of the assumption of the subproof. 4. The premises of the main proof, together with the assumption of the subproof are mutually inconsistent. e) Recall that conjunction is idempotent, that is A ∧ A⇔A and commutative, that is A ∧ B⇔B ∧ A. Which of these properties h old of material conditional (→). 1. Idempotent and commutative 2. Idempotent and not commutative 3. Commutative but not idempotent 4. Neither idempotent nor commutative

{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…

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…

How many ways can we get an even sum when two distinguishable dice are rolled ?

How many three digit numbers can be formed using the digits 1,2,3,4,5,6,7,8 and 9?

how many possible telephone numbers are there when there are seven digits the first two of which between 2 and 9 inclusive the third digit between 1 and 9 inclusive and each of the remaining may be between 0 and 9 inclusive ?

how many 2-digit or 3-digits numbers can be formed using the digits 1,2,3,4,5,6,8 and 9 if no repetition is allowed ?

how many non-negative integers less than 104 that contain the digit 2 ?

what is the numbers of ways to order the 26 letters of the alphabet so that no two of the vowels a,e,i,o,u occur consecutively ?

Determine the truth value of the following statement. (∃n ∈Z)(n2= 3)

Write the statement "The apartment is hot or the air-conditioned is not working, if and only if the temperature is 90°F" in symbolic form if p:the temperature is 90°F q:The air-conditioned is not working r:The apartment is h…

Suppose that the domain A of P(x) is the finite set {1, 2, 3, 4}. Write out the following proposition using disjunctions, conjunctions, and negations. ¬((∀x ∈A)P(x))

(b) Partition A = {1,2,3,4,5,6,7} rvith the minsets generated by Br = {.2,4,6} and B2 = {1,4,5} and also find out hou' manl different subsets of A can )rou generate lrom B. and B2? Expert's answer

Check if the binary relation R defined over Z such that (x, y) ∈ R if and only if x − y is divisible by 4 is an equivalence relation. Ex

Check if f : R 7→ R defined as f(x) = √ x is a function

Which of the following sentences is a proposition whose truth value is T? Select one: a. x + 2 = 11 b. x + y = y + x for every pair of real numbers x and y c. Answer this question carefully. d. Manila is in Cagayan.

How many seven-digit numbers with out repetition of digit are there such that the digit are all non-zero and 5 and 6 do not appear consecutively in either order ?

In how many ways can a committee of 5 teachers and 4 students be chosen from 9 teachers and 15 students ?

How many strings of length n can be formed from the English alphabet ?

How many outcomes are obtained from rollings n indistinguishable dice ?

A student has three mangos two papayas and two kiwifruits .if the student eats one piece of fruit each day and only the type of fruits matters ,in how many different ways can these fruits be consumed ?

A children watches TV at least one hours each days for seven weeks but never more than 11 hours in any week. prove that there is some period of consecutive days during in wich the children watches exactly 20 hours Of tv (if is assumed…

Determine the cofficent of X5Y10Z5W5 in (x-7y+3z-25)25

Use a proof by contraposition to show that if x + y > 2, then x > 1 or y > 1

You cannot edit a protected Wikipedia entry unless you are an administrator. Express your answer in terms of e:“You can edit a protected Wikipedia entry” and a: “You are an administrator.”

Translate the given statements into propositional logic using the propositions provided You can see the movie only if you are over 18 years old or you have the permission of a parent. Express your answer in terms of m: “You can se…

Translate the given statements into propositional logic using the propositions provided You can graduate only if you have completed the requirements of your major and you do not owe money to the university and you do not have an o…

Translate the given statements into propositional logic using the propositions provided To use the wireless network in the airport you must pay the daily fee unless you are a subscriber to the service. Express your answer in terms…

Translate the given statements into propositional logic using the propositions provided You can upgrade your operating system only if you have a 32-bit processor running at 1 GHz or faster, at least 1 GB RAM, and 16 GB free hard d…

~p →~q ~p v {p^q) can get your best answer for this?? Please answer this question.

For each of the following sets, determine whether{ 2} is an element of that set. a) {x ∈ R | x is an integer greater than 1} b) {x ∈ R | x is the square of an integer} c) {2,{2}} d) {{2},{{2}}} e) {{2},{2,{2}}} f ) {{{2}}}

1.construct the truth table for the following compound propositions. Assume all variables denote propositions.A. (pVq) ^ [(~p^q)]B. ~(p—>(q—>(p^q)))C. (p<—>q) V ((~p)—>q)D. (p<—>q) V ((~p)—>(~r))E. (p—>(q—>r))—>((p^q)—>r)

If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on,”, ”If the sailing race is held, then the trophy will be awarded,” and ”The trophy was not awarded” imply the co…

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