Solved: [{1,2,3,4,5}, <=], Draw its Hasse Diagram.

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. …

Show that -p → (q + r) and q→ (p V r) are logically equivalent.

determine the truth value of each of the following truth values if 2+3=5 then 2x3=6

Let p and q be the propositions p : I bought a lottery ticket this week. q : I won the million dollar jackpot. Express each of these propositions as an English sentence. a) ¬p b) p ∨ q c) p → q d) p ∧ q e) p ↔ q f ) ¬p…

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

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.

Suppose that Q(x) is “x+1=2x”, where x is a real number. Find the truth value of the following statement: a) Q(2) b) ∀𝑄(𝑥) c) ∃𝑄(𝑥)

1. Which of the following statements are true and which are false? Give reasons for your answer. (20) i) ‘x 2 +y 2 −3 is not dividible by 4.’ is mathematical statement. ii) The number of onto functions from {1,2,3,4,5,6} to {a,b,…

2. a) Consider the propositions ‘2+3 = 5’ and ’The Sun rises in the West’. i) Write the disjunction of the statements and give its truth value. ii) Write the conjunction of the statements and give its truth value. iii) Write the ex…

2b) Prove or disprove the following statement. (2) “If m divides a n −b n , then m divides abn −ban also.”

2c) Show that any tree with exactly two vertices of degree 1 is a path.

2 d) Write down the converse of each of the following statements: (2) i) If p is a prime number and a and b are any two natural numbers and if p divides a or b, then p divides ab. ii) In a triangle 4ABC, if AB2 +AC2 = BC2 , then ∠…

2. a) Consider the propositions ‘2+3 = 5’ and ’The Sun rises in the West’. i) Write the disjunction of the statements and give its truth value. ii) Write the conjunction of the statements and give its truth value. iii) Write the ex…

If the solution of the recurrence relation αun−1 +βun−2 = f(n),(n ≥ 2) is un = 1−2n+3.2 n , then determine the values of α,β and f(n).

A bank pays you 4.5% interest per year. In addition, you receive |100 as bonus at the end of the year (after the interest is paid). Find a recurrence for the amount of money after n years if you invest |2000.

If 5 points are chosen in a square of side 2cm, show that there will always be two points at a distance of at most √ 2cm.

A die is rolled twice and the sum of the numbers that appear is observed. What is the probability that the sum is either a perfect square or a perfect cube?

Write the expression x1 ∨x2 ∧x3 ∨x4 in conjunction normal form and disjunctive normal form

Find the general form of the solution to a linear homogeneous recurrence relation with constant coefficients for which the characteristic roots are 1,−2 and 3 with multiplicities 2,1 and 2, respectively. The relation also has a non-…

If a planar graph has the degree sequence (2,2,2,3,4,4,5), how many faces will it have? Draw a planar graph with this degree sequence and the number of faces obtained to check your answer

How many numbers form 0 to 999 (0 and 999 inclusive) are indivisible by 7 or 11?

From a survey of 120 people, the following data was obtained: 90 owned a car, 35 owned a computer, 40 owned a house, 32 owned a car and a house, 21 owned a house and a computer, 26 owned a car and a computer, 17 owned all the three…

8. a) Find the generating function of the recurrence an = 6an−1 −5an−2 +1 with initial conditions a0 = 2,a1 = 5.

Express 3x 4 +4x 3 +2x 2 +x in terms of [x]4,[x]3,[x]2 and [x].

Draw the Ferrar graph of the following partitions: i) 20 = 9+6+3+2 ii) 28 = 10+9+8+1 Also, write down the conjugate partitions of each of these partitions.

Evaluate C74

Not having the lecture is sufficient for Dr. Boateng to conduct a quiz or students to make presentations. If the students do not make presentations then Dr. Boateng will conduct the quiz. Therefore, whenever we have the lecture,…

Determine the validity of the following argument: Having a strong mathematical background is necessary and sufficient for understanding the concepts in Information Technology. Having a strong mathematical background and understand…

Prove that 2/n4-3 if and only off 4/n2+3

Q1. a) Let 𝑈 = {𝑥: 𝑥 ∈ 𝑍, 1 ≤ 𝑥 ≤ 12}, 𝐴 = {2𝑥: 𝑥 ∈ 𝑈 𝑎𝑛𝑑 𝑥 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4}, 𝐵 = {𝑥: 𝑥 ∈ 𝑈, 𝑥 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 2} 𝑎𝑛𝑑 𝐶 = {𝑥: 𝑥 ∈ 𝑈, 𝑥2 ≤ 16}. i) List the elements belong to the sets A, B and C respectively. (3 marks) ii) Find 𝐶 − (𝐴̅ ∩…

Define a relation on the set S of all strings of letters: two strings are related if you can get one from the other by reversing one pair of adjacent letters. For example, cow ocw but cow woc.

Let n be an integer. Use Definition 1.6 to explain why 2n + 9 is an odd integer 2n + 9 = 2( )+ 1

In a class of 32 students, 18 offers chemistry, 16 offers Physics 22 offer Mathematics. 6 offer all three subjects, 3 offer Chemistry and Physics only and 5 offer only physics only. Each student offers at least one subject. Find the n…

If x=3 then x<2 is a statement or not?

Identify if the following statements are predicate logic. Give a domain of discourse for each propositional function. A. The movie won the Academy Award for the past 2 years. B. 1 + 3 = 4 C. (x+2)2 is a prime number.

Find the number of integers between 1 and 300 both inclusive that are divisible by the following types: (i). at-least one of 3, 5, 7. (ii). 3 and 5 but not 7. (iii). 5 but neither 3 nor 7

For the following relation on the set {x:x∈Ζ and 1 ≤ x ≤ 12}. List (a) the ordered pairs belonging to the relation: (b) the transitive closure R = {(x, y): xy = 9}

Suppose R(x,y) is the predicate “x understands y,” the universe of discourse for x is the set of students in Discrete Class, and the universe of discourse for y is the set of examples in these lecture notes. Translate the following …

Let f: Z®Z be such that f(x) = x +1. Is f invertible? and if it is, what is its inverse?

P(n): n! > 3n for n ≥ 7. What is the Base Step?

Translate these statements into English sentence, where C(x) is "x is a comedian" and F(x) is "x is funny" and the domain consist of all people. a) Ax(C(x) -> F(x)) b) Ex(C(x) ^ F(x)) c) Ex(C(x) -> F(x))

How many bit strings of length 14 contains the following types (I) atmost three is? (ii) an equal number of 0s and 1s (iii) an odd number of 1s

[{1,2,3,4,5}, <=], Draw its Hasse Diagram.

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