Solved: 3. Let P(x), Q(x), and R(x) be the statements “x is a professor,” “x is ignorant,” and “x is vain,” respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), and R(x), where the domain consists of all people a) No professors are ignorant. b) All ignorant people are vain. c) No professors are vain. d) Does (c) follow from (a) and (b)?

If we have two graph G = (V, E) and G0 = (V, E0 ) on the same set V of vertices we call the union G ∪ G0 the graph (V, E ∪ E0 ). Next we have two claim about the chromatic number χ(G ∪ G0 ) of G ∪ G0 .Prove that χ(G∪G0 ) ≤ χ(G)∗χ(G0 )…

An orientation of a graph G = (V, E) is any directed graph G0 = (V, E0 ) arising by replacing each edge {u, v} ∈ E by the directed edge (u, v) or by the directed edge (v, u). Show that for every planar graph there is an orientation su…

Are these system specifications consistent? ”The file system is not locked only if new messages will be queued. If the file system is not locked, then the system is functioning normally, and conversely. For new messages are not queued…

USING MATHEMATICAL INDUCTION P(n)=1+22+2n+........+2n= 2n+1 - 1

A town has two type of people Knave always speak lie and knative always speak truth, you meet a group of six people A, B, C, D, E and F they communicate with you as following, A Said: All are the knative . B Said: Atleast three ar…

A binary operation € on three variables A, B and C define as return true if at least two true return us true then prove that OR operation on three variable is equivalent to which of the following ? A€(B€C) 2) A€(B€¬C) 3) A€(…

How many Combinations of bit strings length 9 have: a) exactly three 0s? b) at least seven 1s?

how many three letter words can be formed by the word letter EQUATION

In a class of 200 students,70 offered physics,90 chemistry,100 mathematics while 24 did not offer any of the three subject,23 student offered physics and chemistry,41 chemistry and mathematics while 8 offered all the three subjects.Ho…

d)Out of 300 students taking discrete mathematics ,60 take coffee, 27take cocoa,36 take tea,17 take tea only,47 take chocolate only,7 take chocolate and cocoa,3 take chocolate, tea and cocoa,20 take cocoa only ,2 take tea, coffee and …

Construct the switching network of the given compound statements. Then, construct another switching network of the equivalent statement by applying the laws of logic: 1. p v (~p v q) v (p v ~q) ] ^ ~q 2. (p → q) ^ (p ↔ q)

Simplify the following expressions using laws of logic and put what law of logic did you use or apply. p v ~(~p --> q) [(p --> q)^ ~q] --> ~p [(p v q) ^ (p --> ~r) ^ r ] --> q (p v ~q) ^ (p v q) 5. ~[p --> ~(p ^ q)]

Let S be the following statement. “If it is raining, then the ground is wet.” Translate the S into symbol State in English the converse of S State in English the contrapositive of S

Simplify the following expressions using laws of logic: p v ~(~p --> q) [(p --> q)^ ~q] --> ~p [(p v q) ^ (p --> ~r) ^ r ] --> q (p v ~q) ^ (p v q) 5. ~[p --> ~(p ^ q)]

Construct two different assignments that are functions. The functions should be represented by means of ordered pairs graphically. Construct domain, codomain and range(image) of each function.

Writing set into a form of {x | P(x) } 1. {2,4,6,8,10} 2. {a,e,i,o,u} 3. {1,8,27,64,125} 4. {-2,-1,0,1,2} If A = {3,7,12} Find: 5. |A| 6. P(A) 7. | P(A) |

Yesterday, Allan, Berto, and Carlo are playing guess that day. Berto and Carlo gives Allan clues and fake clues. Berto said today is Friday, but Carlo said its Saturday. Allan asked what day is it tomorrow? Carlo said its Monday while…

Determine whether each of these functions is a bijection from R to R. f(x)=2x+1 f(x)=x^2+1

Write the set in the form { x | P(x) } 1. {2, 4, 6, 8, 10}

. Identify the contrapositive of the following statement. If x = 5, then x + 8 = 13

Prove the validity of the arguments using rules of inference. { (p→q) ^ [ q → (r ^ s) ] ^ (p ^ w) ^ [~r v (~w v u) ] } → u

List the members of these sets: (a) A={ x∣x is a real number ∋ x2=1} (b) B={ x∣x is a positive integer less than 100 } (c) C={ x∣x is the square of an integer and x <100 } (d) D={ x∣x is an integer ∋ x2=2 }

Show that {1, 2, 3, 4, . . .} and {2, 4, 6, 8, . . .} have the same cardinality. Hint: find a mapping and show it is 1–1 and onto.

1. A. (20p) Give a Big-O estimate that is as good as possible for each of the following functions. i. (n^3 + n)(logn + n) ii. (n + 1000)(logn + 1) B. (15p) If these are complexities of algorithms that solve the same problem, which …

2. Following algorithm calculates the sum of first n nonnegative integers. Prove the algorithm to be correct using mathematical induction. procedure sum(n: nonnegative integer) if n = 0 then return 0 else return n + sum(n - 1) {o…

In set using venn diagram: The are 60 sciences students in ss3, in a secondary school, 35 of whom study chemistry, 30 study technical drawing, 12 out of those students study biology and chemistry but not technical drawing, 10 study ch…

By using mathematical induction prove that (n+1)! > 2^(n+1) for n, where n is a positive integer greater than or equal to 4

Check the validity of the following argument: “If I study then I will not fail in Mathematics. If I do not play Basket ball then I will study. I failed in Mathematics. Therefore I must have played Basketball

2. For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended. Explain your answer. a) Experience with C++ or Python is required. b) Lunch includes soup or salad. c) Publish or perish. d) To e…

By using mathematical induction prove that (n+1)! > 2^(n+1) for n, where n is a positive integer greater than or equal to 4

In a school, students must take at least one of these subjects: Maths, Physics or Chemistry. In a group of 50 students, 7 take all three subjects, 9 take Physics and Chemistry only, 8 take Maths and Physics only and 5 take Maths and C…

solve these non-homogeneous recurrence relation an+ 4an-1 − 5an-2=n+2 where a0=1 & a1=-1

I am having trouble with the following problem. I am unsure on how to even start. Use propositional logic to prove that the following argument are valid: (A⟶B) ⋀ (A⟶(B⟶C)) ⟶ (A⟶C)

A debating team consists of three boys and two girls. Find the number n of ways they can sit in a row if the boys and girls are each to sit together.

Write the sequence of the following explicit formula: 1. sn = (-4)n , 1 <_ n 2. tn = 92 -5n, 1<_n <_5

1. Blue taxi inc. charges $7.50 for the first 4 miles and 1 dollar for each additional miles. The table shows the cost of travelling from 4 to 10 miles. Mileage Cost 4. 7.50 5. 8.50 6. 9.50 7. 10.50 8. 11. 50 9. 12.50. 10. 13.…

Write the first 4 terms of the following formula, start with n = 1. an = 5n nbn = 3!bn = 3n bn = 3n^2 +2n - 6 gn = 1 × 2 × ... n c1 = 2.5, cn = cn-1 + 1.5 d1 = -3, dn = -2dn-1 + 1

Write the formula of the following sequence, identify if explicit or recursive formula or both. 1. 1.3,5.7 2. 1, -1, 1, -1 3. 1, 4, 7, 10, 13, 16 4. 0, 3, 8, 15, 24, 25 5. 0, 2, 0, 2, 0, 2 ... 6.1, 1/2, 1/4, 1/8, 1/16 7. 0, 1, …

Write the sequence of the following explicit formula. 1. sn = (-4)^n, 1 ≤ n 2. bn = 92 - 5n, 1 ≤ n ≤ 5 3. Blue taxi inc. charges $7.50 for the first 4 miles and 1 dollar for each additional miles. The table shows the cost of travel…

Let S denote the set of all companies listed on the Botswana Stock Exchange. Define the following sets as: A = x | x is in the mining sector , B = x | x has an annual turnover exceeding P10 million , C = x | x has a financial ye…

If A is a set, let A¯ denote the complement of A. Show the following (a) (A ∩ B¯) ⊂ A ∩ B (b) A − B = A if and only if B − A = B

Use bit strings to find the union, intersection, A and B compliments of these sets. The bit strings for the sets {2, 3, 6, 7 and 8} and {1, 3, 5, 7, 9} are 0110011100 and 1010101010, respectively.

Express each of the 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 words “It is not the c…

Find the power set of the set {Ø , { Ø }, a, b, c} where a, b and c are distinct elements.

I. p is "x< 50"; q is "x> 40". Express the following compound propositions as English sentences in as natural a way as you can (a) ¬p (b) ¬q (d) p∨q (e) ¬pV q (f) ¬pV ¬q

2-2.7+2.7^2-...+2(-7)^n = (1- (-7)^n+1)/4

If 11 students offer Biology, 13 students offer Maths, 14 students offer Agric, 9 students offer Maths and Biology, 3 students offer Maths and Agric, 7 students offer Maths, Biology and Agric. How many students offer Biology and Agric?

Let p, q, and r be the propositions. p : You get an A on the final exam. q : You do every exercise in this book. r : You get an A in this class. Write these propositions using p, q, and r and logical connectives (including negations).…

Let A, B, and C are not empty sets, then using Venn diagram represent the following: (i) (A ∩ B) ∩ Cc (ii) Ac ∪ (B ∪ C) (iii)(A – B) ∩ C (iv) (A ∩ Bc) ∪ Cc

Use a set builder notation to give a description of each of these sets. [2 points] a. {0, 3, 6, 9, 12} b. {-3, -2, -1, 0, 1, 2, 3}

Let A = {a, b, c}, B = {x, y} and C = {0, 1}. Find a. C x B X A b. 𝐴

Q(X,y):X+y >or=7 then what is truth value of∀x∃yQ(x,y)

Solve using Polya’s four-step problem solving strategies. 2. According to a student survey on keyboard light preference, 44% liked white, 26% liked blue and 30% liked green. 8% liked both white and green, 12% liked both white and blu…

Determine whether each of the following statements is a proposition or not. If it is, give its truth value.p: Mindanao is an island in the Philippines. q: Find a number which divides your age. r: My seatmate will get a perfect score…

How many 3 letter words can be formed by the word letter EQUATION

Find the formula for a given sequence 3/6, 8/25, 15/62, 24/123, 35/214

Prove by mathematical induction 2n>(n+1)! For all integars n>=2

f:X→Y where X,Y ∈ Z+ , and 0≥ X ≤23 and, 0≥ Y≤59, X is hour of clock and Y is minute of digital clock, Proof the following a. Is f:X→Y is well defined? b. Is f:X→Y a one-to-one correspondence

A f:X→Y where X,Y ∈ Z+ , and 0≤ X ≤23 and, 0≤Y≤59, X is hour of clock and Y is minute of digital clock, Proof the following a. Is f:X→Y is well defined b. Is f:X→Y a on…

1. Determine whether each of the following statements is a proposition or not. If it is, give its truth value. • p: Mindanao is an island in the Philippines. • q: Find a number which divides your age. • r: My seatmate will get a pe…

Determine whether each of the following statements is a proposition or not. If it is, give its truth value. p: Mindanao is an island in the Philippines. q: Find a number which divides your age. r: My seatmate will get a perfect scor…

Solve an+2 = 4an+1 + 21an; n 0 and a0 = 3; a1 = 4

In a class of 16 students, there are 6 male students and 10 female students. In how many ways can you select a group of 5 students from this class such that 3 of the 5 students are males and the other two are females?

Let p, q, and r be the propositions p: You have the flu. q: You miss the final examination. r: You pass the course. Express each of these propositions as an English sentence.

Translate the following argument and use a short truth table to determine whether or not the argument is valid. Either scientists don't know what they are talking about, or the sun will eventually burn out and Earth will become d…

Translate the following sentences and then use a short truth table to determine if the argument is valid. If education was supported but taxes were not raised, then John was elected. John was not elected although education was sup…

Use a short truth table to determine whether or not the following argument is valid. R -> (M v ~C) (P v U) -> C M -> ~P R -> U Thus, (M & U) v ~R Question 4 options: Valid Invalid when C is false, M is false. Invalid …

Use a short truth table to determine whether or not the following argument is valid. ~S -> ~(Q v G) (Q v S) & (G v ~N) (N v ~S) -> L So, S & ~N Question 3 options: Valid Invalid when L is true, S is false. Invalid when…

Use a short truth table to determine whether or not the following argument is valid. E -> J B -> Q D -> (J & ~Q) Thus, (E & B) -> D Question 2 options: Valid Invalid, with J is false, Q is true. Invalid, with D…

Use a short truth table to determine whether or not the following argument is valid. (K & ~C) -> ~(P & R) J -> (K & P) A -> (P & R) Thus, (A & J) -> C Question 1 options: Valid Invalid, with R = false, P = true …

Solve an=3an-1-4,a0=1

Determine whether each of the following statements is true or false. State the reason. i) ∅ ∉ ∅ ii) ∅ ⊆ ∅ iii) {∅} ⊆ ∅

A) Let A={1,2,3,4}A={1,2,3,4} and R a relations on A whose matrices is MR = ⎡⎣⎢⎢⎢1010111000101011⎤⎦⎥⎥⎥[1101010011110001], 1) Show that (A,R)(A,R) is a poset (5 pts) 2) Find maximal, minimal…

State the value of x after the statement if P(x) then x:=1 is executed, where P(x) is the statement “x > 1,” Given that thevalue of x when thisstatement is reachedis [3 marks] a) x=0 b) x=1 c) x =2

If the truth value of P ⊕ Q in 2^3 is 0 0 1 1 1 1 0 0 then what is P <=> Q ins 2^3? If the truth value for P v Q (P or Q) in 2^3 is: 1 1 1 1 1 1 0 0 then what is the truth value for P => Q in 2^3? List all the Premises and Conclusion …

Given that D={a, b, c, d} and R is the relation of D that has the matrix D= 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 1 Find the relation R and the digraph of R

A sample poll of 135 voters revealed the following information in favour of three A ,B and C of a certain political party in Abia state. 60 were in favour of B; 23 were in favour of A alone; 17 were in favour of C alone; 10 were sitti…

3. In how many ways can an elf climb a staircase of 15 stairs if each step it takes can cover 1 or 2 or 3 stairs? Explain why your solution is valid. (Hint: Work on fewer number of stairs first and try to generalize.)

Proof the following by using logical equivalences identities. Are these system specifications consistent by using Reasoning Method? a) ¬(p ∧ (p → ¬q))→¬p b) ¬(q →¬p)→¬q

let Q( x,y) denote the statement 'x=y+5' , what is the truth value of Q(3,4)

Question 1 Let p: I get the job q: I work hard r: I get promoted (a) Write the following proposition in terms of p, q, r and logical connectives. “If I get the job and work hard, then I will get promoted.” (2 marks) (b) Use your…

In a high school 155 students are surveyed and ask which of the foreign languages they learn.45 learn spanish ,56 learn french,61 learn Chinese.22 learn spanish and french,17 learn spanish and Chinese while 24 learn french and Chinese…

a. Let A and B be sets as follows, A= {1,2,3,5,7} and B = {1,5,6,8}. Find: i. A∩B ii. | A∩B| iii. P(A∩B) iv. P(A) and P(B) v. | P(A) | and | P(B) | vi. The Cartesian product…

a. Find the recursive definition for the sequence i. 2 , 4 , 16 , 256 , ...... ii. 1 , 5 , 52 , 53 , 54 , ...

Annulment Law on Boolean Algebra. Type it in MS WORD. Include examples and source of your research. Min. of 5 pages

Show, by the use of the truth table/matrix, that the statement (p v q) ∧ [( ¬p) ∧ (¬q)] is a tautology.

Let p and q be propositions. p: You drive over 65 miles per hour. q: You get a speeding ticket. Write these propositions using p and q and logical connectives. You do not drive over 65 miles per hour. You will get a speeding ticket i…

a. Let A and B and C be sets, prove that A∩(BUC) = (A∩B)U( A∩C).

define the relation on A={1,2,3} such that aRea of a=b+1

define the relation on A={1,2,3} such that aRea of a=b+1 and find it's domain and range

Using quine mccluskey to simplify the sum of product expansion wxyz'+wx'yz+wx'yz'+w'xyz+w'xy'z + w'x'yz + w'x'y'z

1. Given the following: • g: "You can graduate. • m: 'You owe money to the college." . r: "You have completed the requirements of your major." • b: "You have an overdue book." Translate "You…

In any case, if you somehow could not attend the Discrete Mathematics final exam then you will not be able to pass out the course and if you have the fever then you will not be able to pass out the course Construct the proposition by …

Let P(x), Q(x) and R(x) are propositions and the universe of discourse; all students; P(x) = x is intelligent Q(x) = x has solved the problem R(x) = x has clearly understood the clues

Let A, B and C are sets. Prove that ≤ is transitive. (5 marks)

Suppose that 40 Malaysian Chinese are surveyed and they speak at least one of the three dialects, Cantonese, Hokkien or Hakka. It is found that 30 of them speak Cantonese, 20 speak Hokkien and 15 speak Hakka. It is also found that 3…

Simplify ((A U B') n C) U(A'n B)' and write the dual of the result.

Solve (P ∧ Q ∧ R) ∨ (¬P ∧ R ∧ Q) ∨ (¬P ∧ ¬Q ∧ ¬R

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