Solved: QUESTION 4 Below are a number of expressions. State which are terms, some are atomic wffs (well-formed formulae) and some are neither. a) Tet(y) b) Logician(john) c) father_of(quinn) d) 2 + y = z2 e) Angry(x; y;2:00)

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…

a) let us define the following: 1. A “vowel group” is a vowel, v \in∈ {‘a’, ‘e’, ‘i’, ‘o’, ‘u’}, or a concatenation of “two” vowels. 2. A “vowel syllable” is a vowel group. 3. A “consonant syllable” is a “one”, or “two” or at…

a) Consider the function, bin, that converts an integer to a binary digit as: bin : Z 6{-1, 0, 1}, bin(x) = x congruence modulo 2. Now, define the logical NOT function as: NOT: Z\to→ {0, 1}, NOT(x)=\begin{Bmatrix} 0, if…

a) Prove that the set { \downarrow } is an adequate set of connectives. ............................................................................................................................ \ b) Consider the set F =…

Consider the statement form (P \downarrow Q) \downarrow R Now, find a restricted statement form logically equivalent to it, in a) Disjunctive normal form (DNF). b) Conjunctive normal form (CNF).

Consider the human race. Suppose a person x is related to another human being y if x has the same father as y. a) Prove that this is an equivalence relation. b) Compute C(Thabane) and C(Mosisili) c) Can C(Thabane) and C(Mosisi…

Design and implement a minimal 5 modulo up counter. It counts from 0 to 4 and repeats. Design the circuit such that, if the counter enters into the unwanted states: 5, 6 and 7, it should jump into state 0 on the next clock pulse.

a) Let \mathbb{N} be the set of natural numbers, O, the set of odd numbers and S, the set of square numbers. Consider the bijections: f : \mathbb{N} \to→ O, f(n) = 2n + 1 g : O S, g(d)=\frac{d2 -2d +1}{4} Con…

a) Write a finite state machine (FSM) for an automatic reversible 6 modulo counter as follows: The counter counts 0, 1, 2, 3, 4, 5 (when its internal memory input is registering 0) and reverses: 5 4, 3, 2, 1, 0 (when its internal mem…

In the grammar of natural languages, such as Sesotho, noun words are categorised into equivalent noun classes, based on their generating “stems”. A stem is a noun word prefix, which is the “first syllable” of the noun word, from where…

R1={(a, b) € R |a >b|}, the "greater than relation. R2=(a, b) € R|a>=b|}. the "greater than or equal to relation R3=(a, b) €R |a < b), the "less than" relation. R4=(a, b). the "less than or equal to …

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

3 + 5 = 8 is a proposition for p, find the corresponding ῀p.

List the elements of {1, 2, 3, 4} ∩ {2, 3, 5, 7}

Let X = {1, 2, 3, 4, 6, 8, 12, 24} and R be a division relation defined on X. Find the Hasse diagram of the poset.

Prove that the set G={0,1,2,3,4,5} is an abelian group with respect to the multiplication modulo 6.

Consider the following list of numbers: 1 7 8 14 20 42 55 67 78 101 112 122 170 179 190 Apply binary algorithms in order to find the target which are number 42 and number 82. Show your step and calculate the total number of co…

Construct the truth table of the following proposition. 1. (p∧q)→(p∨q) 2. ~(p→q) 3. (p∨q)∧~p

Construct the Combinatorial Circuit of the given output. (p∨q) 2. (p∧ ~q) (p→~q)→(p∨q)

1.Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0, 1), (1, 1), (1, 2), (2, 0), (2, 2), and (3, 0). Find the a) reflexive closure of R. b) symmetric closure of R.

2. Find the transitive closures of these relations on {1, 2, 3, 4}. a) {(1, 2), (2,1), (2,3), (3,4), (4,1)} b) {(2, 1), (2,3), (3,1), (3,4), (4,1), (4, 3)} c) {(1, 2), (1,3), (1,4), (2,3), (2,4), (3, 4)} d) {(1, 1), (1,4), (2,1),…

5. Which relation on the set {1, 2, 3, 4} is an equivalence relation and contain {(1, 2), (2, 3), (2, 4), (3, 1)}. 6. Find the transitive closures of the relation {(1, 1), (1,4), (2,1), (2,3), (3,1), (3, 2), (3,4), (4, 2)} on the s…

{1,2,3,4,6,9}find maximal and minimal and least and greatest number

In general, a statement involving n variables x1 , x 2 ,…, x n can be denoted by P( x1 , x 2 ,…, x n ) and is the value of the predicate P at the n-tuple (x1 , x2 ,…, x n ) . (a) Suppose Q(x , y) denotes the statement “ x=y+3 .” What …

In general, a statement involving n variables x1 , x 2 ,…, x n can be denoted by P( x1 , x 2,…, x n ) and is the value of the predicate P at the n-tuple (x1 , x2 ,…, x n ). (a) Suppose Q(x , y) denotes the statement “ x=y+3 .” Wha…

ind the smallest relation containing the relation {(1, 2), (1, 4), (3 , 3), (4, 1)} that is a) reflexive and transitive. b) symmetric and transitive. c) reflexive, symmetric, and transitive.

Prove by contrapositive that if n = a*b, where a and b are positive integers, then a ≤ √n or b ≤ √n

Q.1 Prove by contrapositive that if n = a*b, where a and b are positive integers, then a ≤ √n or b ≤ √n

If 4 cards are selected from a standard 52- card deck must be at least 2 be of the same suit.Why?

Are these system specifications consistent? “The system is in multiuser state if and only if it is operating normally. If the system is operating normally, the kernel is func- tioning. The kernel is not functioning or the system is…

Are these system specifications consistent? “The router can send packets to the edge system only if it supports the new address space. For the router to support the new ad- dress space it is necessary that the latest software relea…

Are these system specifications consistent? “If the file system is not locked, then new messages will be queued. If the file system is not locked, then the system is func- tioning normally, and conversely. If new messages are not …

When planning a party you want to know whom to invite. Among the people you would like to invite are three touchy friends. You know that if Jasmine attends, she will become unhappy if Samir is there, Samir will attend only if Kanti wi…

SHOW that every open interval is uncountable

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

{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…

{F} 2. Find the transitive closures of these relations on {1, 2, 3, 4}. a) {(1, 2), (2,1), (2,3), (3,4), (4,1)} b) {(2, 1), (2,3), (3,1), (3,4), (4,1), (4, 3)} c) {(1, 2), (1,3), (1,4), (2,3), (2,4), (3, 4)} d) {(1, 1), (1,4), (2…

{F} Construct the Combinatorial Circuit of the given output. (p∨q) 2. (p∧ ~q) (p→~q)→(p∨q)

USE THE METHOD OF DIRECT TO PROVE: If a is an odd integer, then a’2 + 3a +5 is odd.

This question has 2 parts. Part 1: Suppose that F and X are events from a common sample space with P(F) 6= 0 and P(X) 6= 0. (a) Prove that P(X) = P(X|F)P(F) + P(X|F¯)P(F¯). Hint: Explain why P(X|F)P(F) = P(X ∩ F) is another way of wri…

Let p and q be the following propositions: p: Angela misses the final exam. q : Angela passes the course. Write the following propositions using p, q and negations and logical connectives: (a) Angela either m…

250 members of a certain society have voted to elect a new chairman. Each member may vote for either one or two candidates. The candidate elected is the one who polls most votes. Three candidates x, y z stood for election and when t…

QUESTION 4 Below are a number of expressions. State which are terms, some are atomic wffs (well-formed formulae) and some are neither. a) Tet(y) b) Logician(john) c) father_of(quinn) d) 2 + y = z2 e) Angry(x; y;2:00)

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