Solved: Let C={A1, A2, ..., An} be a collection of finite sets that are pairwise disjoint. Further suppose that |Ai|=i. Compute |U(i=1 to n)Ai|, and write your answer in the simplest closed form possible.

Draw a simple, undirected graph yourself, the vertices are connected with each other including 8 vertices and 14 edges. Find the shortest path from two arbitrary vertices: a) The weight of each edge is 1. b) Self-weighting for…

1. For the following functions, provide the domain and range a) f: . For each x ∈ domain f(x) = cx. b) g: . For each x,y ∈ domain, g(x,y) = yxd.

Are the following functions from R to R? If not, explain why a) f(x) = 1/x b) f(x) = 2x + 7

2. For each function below (from Z to Z), indicate whether the function is onto, one-to-one, neither or both. If the function is not onto or not one-to-one, give an example showing why. If the function is bijective, find and show …

2. Consider three functions f, g, and h (from R to R+). Let f (x) = x2, g(x) = 2x and h(x) = log2(x) a) Evaluate f ◦ g(0) b) Evaluate h ◦ g(5) c) Evaluate h ◦ f(0) d) Give a mathematical expression for h ◦ g e) Give a ma…

2. Indicate if the following statements are possible. Justify your answer. If the answer is “yes”, give a specific example of the functions. Let f: and g: Z be two functions. Use a diagram if that helps explain the function. a)…

2. Provide the first 5 terms of the following sequence, excluding any initially provided terms. Also, indicate which sequences are increasing, decreasing, non-increasing, and/or non-decreasing. a) The nth term is ⌈⌉. b) The nt…

2. Let S = { :(a1a2a3a4∈ N and 0 ≤ ≤ 9 for each i = 1, 2, 3, 4}. In other words, S is the set of all 4-digit strings with each digit between 0 an 9. (a) Show that the function f : S → S deﬁned by f(a1a2a3a4) =a4a3a2a1 is a bij…

State whether or not the following functions have a well-deﬁned inverse. If the inverse is well-deﬁned, deﬁne it. If it is not well-deﬁned, provide justiﬁcation. a) f : Z → Z. f(x) = 7x – 7 b) f : R → R. f(x) = 7x – 7 c) A…

For each list of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assume the sequence starts with . Do not forget initial conditions if required. a) 7, 11…

List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3}, where (a, b) ∈ R if and only if a) a = b. b) a + b = 4. c) a > b. d) a ∣ b. e) gcd(a, b) = 1. f ) lcm(a, b) = 2.

Show that A ⊕ B = (A ∪ B) - (A ∩ B).

For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)…

Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if a) a is taller than b. b) a and b were born on the same d…

Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if a) everyone who has visited Web page a has also visited Web pa…

a) not reflexive: x = x symmetric: if x ≠ y then y ≠ x not antisymmetric: if x ≠ y then y ≠ x not transitive: for example, if 5 ≠ 6 and 6 ≠ 5 then 5=5 b) not reflexive: for example, 0\cdot0<10⋅0<1 Is symmetric because we…

Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if a) x ≠ y. b) xy ≥ 1. c) x = y + 1 or x = y − 1. d) x ≡ y (mod 7). e)…

Answer these questions for the poset ({3, 5, 9, 15, 24, 45}, |). a) Find the maximal elements. b) Find the minimal elements. c) Is there a greatest element? d) Is there a least element? e) Find all upper bounds of {3, 5…

Let R1 and R2 be the “congruent modulo 3” and the “congruent modulo 4” relations, respectively, on the set of integers. That is, R1 = {(a, b) | a ≡ b (mod 3)} and R2 = {(a, b) | a ≡ b (mod 4)}. Find a) R1 ∪ R2 b) R1 ∩ R2 c) R…

Represent each of these relations on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order). a) {(1, 1), (1, 2), (1, 3)} b) {(1, 2), (2, 1), (2, 2), (3, 3)} c) {(1, 1), (1, 2), (1, 3), (2, 2), (…

Which of these relations on the set of all functions from Z to Z are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) {(f, g) | f (1) = g(1)} b) {(f, g) | f (0) = …

Show that the relation R consisting of all pairs(x, y) such that x and y are bit strings of length three or more that agree in their first three bits is an equivalence relation on the set of all bit strings of length three o…

Determine whether function f : Z × Z → Z is onto if (i) f(m,n)=2m−n (ii) f(m,n)=m+n+1

Suppose that a password for a computer system must have at least 6, but no more than 9 characters, where each character in the password is a lowercase English letter, or an uppercase English letter, or a digit, or one of the five spec…

Solve the linear congruence 34x ≡ 53(mod 89).

3. Which of the intervals (0, 5), (0, 5], [0, 5), [0, 51, (1,4], [2, 3], (2, 3) contains a) 0? b)1?c)2? d) 3? e)4? f)5?

Let P(x,y) denote the sentence x2 + 1≥ x + 1. What are the truth value of the following where the domain of x and y is the set of all integers? a. ⱯxⱯyP(x,y) b. ⱯxƎyP(x,y) c. ƎxⱯyP(x,y) . d. ƎxƎyP(x,y)

use the Euclidean algorithm to express gcd(94, 159) as a linear combination of 94 and 159

Suppose that the domain of the predicateP(x) consists of −2, −1, 0, 1, and2. Write out each of the following predicate logic formulas in propositional logic formulas using disjunctions, conjunctions, negations, or their combinations.

Construct a truth table for each of these compound propositions. [6 marks] a) p ⊕ p b) p ⊕￢p c) p ⊕￢q d) ￢p ⊕￢q e) (p ⊕ q) ∨ (p ⊕￢q) f ) (p ⊕ q) ∧ (p ⊕￢q)

4. (2 points) Solve these recurrence relations together with the initial conditions given. Each is worth 1 point. a) an = 6an-1 - 11an-3 +6an-3 for n > 3, a0 = 4, a1 = 6, a2 = 12 b) an = an–1 +9an-2 – 9an-3 for n > 3, a0 = 6, a1= 0,…

Let x and y be real numbers. Prove that, if 5x+y>11, then x>2 or y>1.

Determine the minimum number of separate racks needed to store the chemicals given in the table (1st column) by considering their incompatibility u sing graph coloring technique. Clearly state you steps and graphs used. Chemical…

Write A= {10, 20, 30, 40, 50} using rule method.

Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function. a) the function that assigns to each bit string the number of ones in the string …

1. Expand the following using the Binomial Theorem a. (x-3y)^3 b. (2a+b^-3)^4 c. (a+b+c)² 2. Solve the recurrence relation Pn = Pn-1 + n with the initial condition p_1 = 2 using iteration.

P = I pass this class G = I go to class every day. H = I do all the homework exercises. Translate the following sentences into propositional logic. (1) Students will not pass this class unless they go to class every day and do all…

Solve the following recurrence relations for the initial conditions given. The population of Utopia increases 5 percent per year. In 2000 the population was 10,000. What was the population in 1970?

Write the negation of the following statement: ∀𝑥∃𝑦(𝑥 + 𝑦 = 2 ∧ 2𝑥 − 𝑦 = 1

Suppose that the domain of the predicateP(x) consists of −2, −1, 0, 1, and2. Write out each of the following predicate logic formulas in propositional logic formulas using disjunctions, conjunctions, negations, or their combinations.

illustrate the venn diagram for the sets A,B,and C in the U={1,2,3,4,5,6,7,8} a={1,2,3,4,5,6,7}, b={1,5,6,7}, c={1,2,3,6}

Q1: Let A, B, C be three sets. Prove that (𝐴 − 𝐵) − 𝐶 = (𝐴 − 𝐶) − (𝐵 − 𝐶) using following proof methods: a) Membership table b) Set builder notation and propositional logic Q2 : Find generalized union and intersection as de…

Draw Hasse diagram repressing the partial order on {(a,b) : a\b } on {1,2,3,4,6,8,12}

I. Create a truth table for all of the 11 Logical Equivalence. II. Find if the following are logically equivalent 1.(~p v q) ^ (~q) <=> ~(p v q) 2.(p ^ ~q) v (~p v q) <=> T

29. Prove or disprove that if m and n are integers such that mn = 1 then either m = 1 and n = 1 , or else m = - 1 and n = - 1 .

Let A and B be subsets of a universal set U. Show that A ⊆ B if and only if B ⊆ A.

Use rules of inference to show that the hypotheses ”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 awa…

6. (a) Draw the logic circuit for the following expression: AB + A(B+C) (b) Simplify the expression in 6(a) by using the rules of Boolean algebra provided in (5). (c) Draw the simplified logic gate circuit derived in (b)

State the Pigeonhole Principle. Prove that if six integers are selected from the set [3,4,5,6,7,8,9,10,11,12] there must be two integer whose sum is fifteen.

A computer randomly chooses a binary string of length 6. What is the probability that exactly two characters in the string turn out to be 1s?

State the Pigeonhole Principle. Prove that if six integers are selected from the set [3,4,5,6,7,8,9,10,11,12] there must be two integer whose sum is fifteen.

A box contains 6 red, 8 green,10 black 11 yellow and 12 white balls. What is the minimum number of balls we have select from box to guarantee that 9 balls are of the same colours.

Prove that if 10 points are placed in a 3cm by 3cm squre board than two points must be at least √2 cm apart.

6. (a) Draw the logic circuit for the following expression: AB + A(B+C) (b) Simplify the expression in 6(a) by using the rules of Boolean algebra provided in (5). (c) Draw the simplified logic gate circuit derived in (b)

a particular algorithm increases in time as the number of operations n increases. Suppose the time complexity of this algorithm is given by: f(n)=4n2+5n2*log(n) Show that f(n) is O(g(n)) for g(n) = n3

State the Pigeonhole Principle. Prove that if six integers are selected from the set [3,4,5,6,7,8,9,10,11,12] there must be two integer whose sum is fifteen.

A weighted voting system for voters A, B, C, D, and E is given by {35: 29, 11, 8, 4, 2}. The weight of voter A is 29, the weight of voter B is 11, the weight of voter C is 8, the weight of voter D is 4, and the weight of voter E is 2.…

A toy shop has 15 airplanes, 15 buses, 17 trains, and 20 bikes in the stock. a. How many ways are there for a person to take 15 toys home if all the airplanes are identical, all the buses are identical, all the trains are identical an…

Let P: It is below freezing and Q: It is snowing. What the symbolic form of the "It is not snowing, it is below freezing".

Prove that in a graph G, the sum of the degrees of the vertices is equal to twice the number of edges. Consequently, the number of vertices with odd degree is even

⋃ 𝐴𝑖 ∝ 1 and ⋂ 𝐴𝑖 ∞ 1 a) 𝐴𝑖 = {𝑖, 𝑖 + 1,𝑖 + 2, … } b) 𝐴𝑖 = {0,𝑖} c) 𝐴𝑖 = {−𝑖, − 𝑖 + 1, … , −1, 0, 1, … , 𝑖 − 1, 𝑖} d) 𝐴𝑖 = {−𝑖, 𝑖}

Check whether the following graphs have an Eulerian and/or Hamiltonian circuit.

Let P (x) denote the statement “x ≤ 4.” What is the truth value of p(6) ?

Find the bitwise “and” 0110 1101 , 1001 0011

Mathematical logic

What and Define Conjunction

What and Define Disjunction of OR and Exclusive

What and Define Negation

What and Define conditional and biconditional operator

What and Define biconditional

What and Define Tautology and contradiction

What and Define Equivalent formula or laws of algebra of propositions

What and define Rules of inferences

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen? How many must be selected to guarantee that at least 3 hearts selected?

During a month with 30 days, a base ball team plays at least one game a day but no more that 45 games. Show that there must be a period of some number of consecutive days during which the team must plays exactly 14 games.

Define group. Show that the set P3 of all permutations on three symbols 1,2,3 is a finite non-abelian group of order six with respect to permutation multiplication as composition.

State the Pigeonhole Principle. In a result sheet of a list of 60 students, each marked “Pass” or “Fail “. There are 35 students pass. Show that there are at least two students pass in the list exactly nine students apart. (for exam…

Suppose that you meet three people Aiman, Borhan, and Camelia. Can you determine what Aiman, Borhan, and Camelia are if Aiman says "All of us are knaves" and Borhan says "Exactly one of us is a knave."?

The recursive algorithm given below can be used to compute gcd(a, b) where a and b are non-negative integer, not both zero. procedure gcd(a, b) if a > b then gcd(a, b) := gcd(b, a) else if a = 0 then gcd(a, b) := b…

7. (a) Use the rules of Boolean algebra provided in (5) to simplify the following logic gate circuit: (b)Draw the simplified logic gate circuit in (a)

Determine whether each following statements about Fibonacci numbers is true or false A. 2Fn >F n+1 for n≥3 B. 2F n+4 = f n+3 for n≥3

Consider a recurrence relation an = -3an-1 + n for n = 1,2,3,4,… with initial conditions a1 = 3. Calculate a3.

Consider a recurrence relation an = an-1 - 3an-2 for n = 1,2,3,4,… with initial conditions a1 = 3 and a2 = 5. Calculate a5.

State the Pigeonhole Principle. In a result sheet of a list of 60 students, each marked “Pass” or “Fail “. There are 35 students pass. Show that there are at least two students pass in the list exactly nine students apart. (for exam…

State the Pigeonhole Principle. A chess player wants to prepare for a championship match by playing some practice games in 77 days. She wants to play at least one game a day but no more than 132 games altogether. Prove that there is…

b) Given p="Front door is open q="Back door is open" r="Light is on" s="The lift doors are open" t="The lift is moving", i) write the follow…

When considering the set of all the natural numbers (ℕ), show whether the mathematical operations of addition, subtraction, multiplication and division are: (a) Associative (b) Commutative.

Given that 𝐺 = {𝑎 ∈ ℝ|𝑎 ≠ −1} and 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 𝑎𝑏, show that (𝐺, ∗) is indeed a group.

You are required to prepare a 15-minute presentation aimed towards new students engaged under a technology company’s graduate scheme. Your presentation will include evidence of: 1. An explanation that adequately explains why group t…

Suppose a recurrence relation an=4an−1−4an−2 where a1=14 and a2=40 can be represented in explicit formula, either as: Formula 1: an=pxn+qnxn or Formula 2: an=pxn+qyn where …

a n −4a n−1 +4a n−2 =n 2 \displaystyle\sum_{n=2}^{\infin}a_nx^n-4\displaystyle\sum_{n=2}^{\infin}a_{n-1}x^{n}+4\displaystyle\sum_{n=2}^{\infin}a_{n-2}x^{n}=\displaystyle…

Let C={A1, A2, ..., An} be a collection of finite sets that are pairwise disjoint. Further suppose that |Ai|=i. Compute |U(i=1 to n)Ai|, and write your answer in the simplest closed form possible.

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