Solved: Consider the following English statements and express as a logical expression with the same meaning. The domain is the set of all students at Hogwarts. H(x): x is a member of the Hufflepuff. R(x): x is a member of Ravenclaw. T(x,y): x takes a class with y. a. There exists a student at Hogwarts b. Every student is taking a class with Harry Potter c. No student was placed in Hufflepuff d. If a student took a class with Ron Weasley, then they did not take one with Harry Potter. e. At least one student in Hufflepuff took a class with a student in Ravenclaw Two students in Hufflepuff took a class with the same Ravenclaw student

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

I. SET. A. List the members of these sets 1. {x | x is a positive integer less than 10} 2. {x | x is an integer such that x2 = 2} B. Consider the following sets. Universal set (U) = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, a, b, c, d}…

Given the following set: 1. X = {1, 2, 3, 4, 5} defined by the rule (x, y) ∈ R if x + y ≤ 6 a. List the elements of R b. Find the domain of R c. Find the range of R d. Draw the digraph e. Properties of the Relation

List the members of the following sets 1. {x| x is real numbers and x2 = 1} 2. {x| x is an integer and -4 < x ≤ 3} B. Use set builder notation to give description of each of these sets. 1. {a, e,i ,o, u} 2. {=2, -1, 0, 1, 2} C. …

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 can …

What is the complete proof for INCLUSION-EXCLUSION FORMULA and BOOLE'S INEQUALITY? Step by Step Solutions. The BOOLE'S INEQUALITY that I meant was both: 𝑃(𝐴1 ∪ 𝐴2 ∪ … ∪ 𝐴𝑛 ) ≤ 𝑃(𝐴1 ) + 𝑃(𝐴2 ) + ⋯ + 𝑃(𝐴𝑛) or 𝑃(⋃ 𝐴𝑖 𝑛 𝑖=1 ) ≤ ∑ 𝑃(𝐴𝑖) 𝑛…

Universal set (U) = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, a, b, c, d} A = {a, b, c} B = {-2, -1, 0, 1, 2} C = {2, 4, 6} find: B-C C x A P(A) |P (B)|

RELATION. Given the following set: 1. X = {1, 2, 3, 4, 5} defined by the rule (x, y) ∈ R if x + y ≤ 6 a. List the elements of R b. Find the domain of R c. Find the range of R d. Draw the digraph e. Properties of the Relation

A. List the members of the following sets 1. {x| x is real numbers and x2 = 1} 2. {x| x is an integer and -4 < x ≤ 3} B. Use set builder notation to give description of each of these sets. 1. {a, e,i ,o, u} 2. {=2, -1, 0, 1, 2} …

Find simpler statement forms that are logically equivalent to p ⊕ p and (p ⊕ p) ⊕ p. b) Is (p ⊕ q) ⊕ r ≡ p ⊕ (q ⊕ r)? Justify your answer. c) Is (p ⊕ q) ∧ r ≡ (p ∧ r) ⊕ (q ∧ r)? Justify your answer.

If p → q is false, can you determine the truth value of (~p)˅(p ↔ q)? Explain your answer. If p → q is true, can you determine the truth value of ~(p → q)˄~p? Explain your answer.

Prove the equivalence of the following in three different ways (truth table, simplification, each is a logical consequence of the other): p → (q ∨ r) ≡ (p ∧ ~q) → r.

Use the properties to verify the logical equivalences in the following. Supply a reason for each step. a. (p ∧∼ q) ∨ p ≡ p b. p ∧ (∼ q ∨ p) ≡ p c. ∼ (p ∨∼ q) ∨ (∼ p ∧∼ q) ≡ ∼ p d. ∼ ((∼ p ∧ q) ∨ (∼ p ∧∼ q)) ∨ (p ∧ q) ≡ p e. (p ∧…

A. List the members of the following sets 1. {x| x is real numbers and x2 = 1} 2. {x| x is an integer and -4 < x ≤ 3} B. Use set builder notation to give description of each of these sets. 1. {a, e,i ,o, u} 2. {=2, -1, 0, 1, 2} …

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 can …

If 𝑝 → 𝑞 is false, can you determine the truth value of (~𝑝)˅(𝑝 ↔ 𝑞)? Explain your answer. If 𝑝 → 𝑞 is true, can you determine the truth value of ~(𝑝 → 𝑞)˄~𝑝? Explain your answer.

a) Find simpler statement forms that are logically equivalent to 𝑝 ⊕ 𝑝 and (𝑝 ⊕ 𝑝) ⊕ 𝑝. b) Is (𝑝 ⊕ 𝑞) ⊕ 𝑟 ≡ 𝑝 ⊕ (𝑞 ⊕ 𝑟)? Justify your answer. c) Is (𝑝 ⊕ 𝑞) ∧ 𝑟 ≡ (𝑝 ∧ 𝑟) ⊕ (𝑞 ∧ 𝑟)? Justify your answer.

Find the graph of the function f(n)=1-n from integers to integer

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 can gra…

X=factors of 30 R=x/y draw hasse diagram

A. Write each of the following sets by listing their elements between braces. 2. {3x+2:xϵZ} 4. {xϵN:-2

A. Write out the indicated sets by listing their elements between braces. 2. Suppose A={π,e,0} and B={0,1} (a) AxB (b) BxA (c) AxA (d) BxB (e) ØxB (f) (AxB)xB (g) Ax(BxB) (h) AxBxB 4. {nϵZ:2

A. List all the subsets of the following sets. 2. {1,2,Ø} 8. {{0,1},{0,1,{2}},{0}} B. Write out the following sets by listing their elements between braces. 12. {X:X ⊆{3,2,a} and │X│=1 C. Decide if the following statement…

A. Find the indicated sets. 2.ℙ({1,2,3,4}) 6. ℙ({1,2})x ℙ({3}) 10. {Xϵ ℙ({1,2,3}):│X│≤1} B. Suppose that │A│=m and │B│=n. Find the following cardinalities. 14. │ ℙ(ℙ(A))│ 18. │ ℙ(Ax ℙ(B))│

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 can …

In how many ways can six boys and four girls be arranged in a straight line so that no two girls are ever together.

Let A = {1, 2, 3, 4, 5, 6, 7} and R = {(x, y) | x –y is divisible by 3} Show that R is an equivalence relation. Draw the graph of R.

A. List the members of the following sets 1. {x| x is real numbers and x2 = 1} 2. {x| x is an integer and -4 < x ≤ 3} B. Use set builder notation to give description of each of these sets. 1. {a, e,i ,o, u} 2. {=2, -1, 0, 1, 2}…

Use Deductive reasoning to solve the following logic puzzle. 1. Four kids went to an unusual pet store. Each child picked out a different animal to take home. Can you match the child with their new friend? Clues: 1. No child has …

given x = [101011110] and y = [110101001] find xoy and x ^ y

Let R be the relation on the set A = {1,2,3,4,5,6,7} defined by the rule (a,b) belongs to R, if the integer is divisible by 4, List the elements of R and its inverse

Show step by step: find the inverse of {(-3,1) (2,4) (8,9)}

List the members of the following sets 1. {x| x is real numbers and x2 = 1} 2. {x| x is an integer and -4 < x ≤ 3} B. Use set builder notation to give description of each of these sets. 1. {a, e,i ,o, u} 2. {=2, -1, 0, 1, 2} C. …

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