Solved: A number of students prepared for an examination in physics, chemistry and mathematics. Out of this number, 15 took physics, 20 took chemistry and 23 took mathematics. 6 took both physics and mathematics and all those who took physics also took chemistry. One student fell ill and failed to write any of the papers. A. Draw a venn diagram for the above information B. Use the venn diagram to find I. How many students took one of the papers II. How many students? prepared for the exams?

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…

whether it is the linearly ordered set or not. Let A = $1, 2, 3, 4, 6, 9} and let R be the relation on A defined by "x divides y " written x/y. a) Write R as a set of ordered pairs b) Drawits directed graph. c) Find ind…

Solution: a) p∧q b)p∧¬q c)¬p∧¬ q d)p∨q e)p→ q f)(p∨q)∧(p→ ¬ q) g)p↔q

Let p and q be the propositions p : It is below freezing. q : It is snowing. Write these propositions using p and q and logical con- nectives (including negations). a) It is below freezing and snowing. b)…

Determine whether each of these compound propositions is satisfiable. a) (p ∨ ¬q) ∧ (¬p ∨ q) ∧ (¬p ∨ ¬q) b) (p → q) ∧ (p → ¬q) ∧ (¬p → q) ∧ (¬p → ¬q) c) (p ↔ q) ∧ (¬p ↔ q)

Explain, without using a truth table, why (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r) is true when at least one of p, q, and r is true and at least one is false, but is false when all three variables have the same truth value. Expert's ans…

Construct the truth table of (p \land \lnot q) \lor \lnot (q \land r) \lor (r \land p) Use truth tables to prove that (p \land \lnot q) \lor \lnot (q \land r) \lor (r…

Let p and q be the propositions “The total amount is discounted” and “The items have been packed,” respectively. Express each of these compound propositions as an English sentence.

What is p ⊕ p ? true or false

a) i) Give an inductive formula for the sum of the first n odd numbers: 1 + 3 + 5 + ... + 2n -1 Show your induction process. ii) Use the proof by mathematical induction to prove the correctness of your inductive formula in i) a…

Consider the following premises: 1. A \to→ (B \to→ A) is a Theorem of Propositional Calculus/Logic (i.e. it’s logically valid), for all statement forms A and B. Suppose then that the following are the temporary axioms…

. 1. Let D = {3, 9, 15}, E = {3, 15, 9}, F = {3, 3, 9, 15, 15, 15}. What are the elements of D, E, and F? How are D, E, and F related? 2. How many elements are in the set {1, {2}, [1, 2}}? 3. For each positive int…

1. Write the following sentences in the form of if... then and then find the inverse, converse, and contrapositive [8] i) You can become successful and happy unless you lead a lazy life. ii) We enter the password or we click on the…

a) The aim is to determine whether the set ∅ is a power set of a set or not. By definition, given a set S, the power set of S is a set of all subsets of the set S. The power set of S is denoted by P(S). The power set of every set in…

Determine whether each of these sets is the power set of a set. where a and b are distinct elements. a) ∅ b) {∅.{∅}} c) {∅, {a}. {∅, a)} d) {∅, {a}. {b}. {a, b}}

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 ¬P(x) d) ∀x ¬P(x)

Let C(x) be the statement “x has a cat,” let D(x) be the statement “x has a dog,” and let F(x) be the statement “x has a ferret.” Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Let…

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. a) No one is perfect. b) Not everyone is perfect. c) All your friends are perfect. d) At least one of your friends is…

Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the…

Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not th…

Determine whether ∀x(P(x) → Q(x)) and ∀xP(x) → ∀xQ(x) are logically equivalent. Justify your answer.

Show that ∃x(P(x) ∨ Q(x)) and ∃xP(x) ∨ ∃xQ(x) are logically equivalent.

Show that ∃xP(x) ∧ ∃xQ(x) and ∃x(P(x) ∧ Q(x)) are not logically equivalent.

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

A Sesotho word cannot begin with of the following letters of alphabet: D, G, V, W, X, Y and Z. We define the relation: A Sesotho word x is related to another Sesotho word y if x begins with the same letter as y. Determine whet…

Let O be the set of odd numbers and O’ = {1, 5, 9, 13, 17, ...} be its subset. Define the bijections, f and g as: f : O \to→ O’, f(d) = 2d - 1, \forall∀ d \in∈ O. g : \NuN \to→ O, g(n) = 2n + 1, \foral…

Consider the following premises: 1.A-->(B-->A) is a theorem of proportional calculus for all statements A and B. 2.suppose then that the following are the temporary axioms. a)w b)y c)y-->z Using the log…

If A={1,3,5,7} and B={1,3,7}. Is set B a proper subset of set A? Explain. Expe

Prove that \sum_{i=0}^{n} 2^{i} = 2^{n + 1} - 1 Use mathematical induction for this proof and discuss/explain each step.

If A={1,2,3}, identify the power set of the given set A. P(A)= P(A)={ } and P(A)=

show that (A ∪ B)\C ⊆ A ∪ (B\C)

Prove that \sum_{i=0}^{n} 2^{i} = 2^{n + 1} - 1 Use mathematical induction for this proof and discuss/explain each step.a) i) b_0=1, b_1=2, b_n=2^nb 0 =1,b 1 =2,b n =2 n Let a_1=b_0=1, a_2=b_1=2, ..., a_{n+1}…

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