Solved: Let a and b be two Natural Numbers, such that the greatest common divisor of a and b is 63, and the least common multiple of a and b is 44452800. If ’b’ is an odd number, what is the minimum value of ’a’ possible? [Hint: a · b = gcd(a, b) · lcm(a, b)]

b. Determine whether each of these functions is a bijection from Z to Z. f (n) = n2 + 1

Prove that ((P Ꚛ Q) →¬R) ↔¬P is a tautology, a contradiction or contingency.

Consider the following two sets A & B: A= {4, 8, 12, 16, … } B = {1, 3, 5, 7, 9, … } Let be a function from z × z to , such that f(m,n) = (m*m)-(n*n). . i) Show that every element of the set A has a preimage under the functio…

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

Express the following using language of Predicate Calculus, where it is understood that the people being discussed are in the courtroom. If any sentence is ambiguous, give all symbolic versions. (i) All judges are sober (ii) There is …

Let f be a function from the set A to the set B. Let S and T be two disjoint subsets of A (i.e S∩T=∅); then which of the following cannot be true: a) if f is invertible, then f(S)∩f(T)=∅ b) if f(S)∩f(T)≠∅, then f is a one-to-one…

3)Let x=n+ε, where n is an integer and 0≤ε<1. If ⌊55x/7⌋=8n, then which of the following cannot be a value of x: a) 14.3 b) 37.7 c) 46.9 d) 51.6 4) Let x=n+ε, where n is an integer and 0≤ε<1. If ⌈33x/8⌉=5n, then which of the fol…

i) Let S={2,4,7} and T={1,3,5}. Find f(S×T) if f(x,y)=⌊14x/3y⌋ Type/Insert your answer here! Note: No partial credit would be admissible in this question f(x,y)=x^2+y^3 Type/Insert your answer here! Note: No partial credit…

Find f∘g and g∘f , where f(x)=x^2+2x+1 and g(x)=x^2-20, are functions from R to R. Type/Insert your answer here!

Let f be a function from Z to R, such that f(x)=x/10, then f is a) an increasing function b) a strictly increasing function c) a decreasing function d) an onto function

In the following argument, determine the validity or otherwise of the Statement: a) “If you aren’t polite, you won’t be treated with respect. You aren’t treated with respect. Therefore, you aren’t polite. b) “If you play football …

1. Suppose that f is defined recursively by: f (0) 5= and f n( + =1) 2fn+5. Find f(1), f(2), f(3) and f(4)?

The study involved 55 students (23 male, 32 female) and used a mixed methods approach, involving a questionnaire with open ended questions, Likert scale questionnaire and interviews that aimed to determine students' perceptions of the…

In a high school, 50 students are surveyed and asked about the interest of the student. 23 students interested in English subject, 14 interested in Mathematics, and 11 in Urdu, 6 is in English an mathematics, 4 is in English and Urdu …

Let S={2, 4, 7} and T={1, 3, 5} . Find f(S×T) if f(x,y)=14x/3y

Consider the following assertions about the sets A, B and C. Write them down in the language of predicate logic. Use only the constructions of predicate logic (∀, ∃, ¬, ⇒, ∧, ∨) and the element of symbol (∈). Do not use derived notion…

1) Let be a function from Z to R, such that , then is a) an increasing function b) a strictly increasing function c) a decreasing function d) an onto function

MATHEMATICAL INDUCTION AND RECURRENCE 1. What is the base case for inequality 3n > n2 , where n = 2? (2 pts) A. 3 > 1 B. 9 > 4 C. 6 > 4 D. 4 < 9 2.For the mathematical induction to be true, what type of number should be the valu…

5. If P(k) = k2 (k + 2)(k – 1) is true, then what is P (k + 1)? A. (k + 1)2 (k + 2)(k) B. (k + 1)2 (k + 2)(k) C. (k + 1)(k + 3)(k) D. (k + 1)2 (k + 3)(k) 6. Using the principle of mathematical induction, 2n-1 is divisible by whic…

How many bit strings of length 11 have more 0s than 1s?

Using the handshaking principle,determine the number edges of a graph with fourteen vertices and each with degree six

Example 1: Let P, Q and R be the propositions P: It’s raining outside. Q: It’s safe to drive. R: The roads are slippery. a) Though it’s raining outside but the roads are not slippery. b) It’s safe to drive if and…

p ∧¬q

Use set builder notation to give a description of each of these sets. a) {3, 6, 9, 12} b) {−4,−3,−2,−1, 0, 1, 2, 3, −4}

(¬p→r) ∧ (q ↔p)

Find all primes less than a specified positive integer n. (let’s say n =100).

Construct a truth table for each of these compound propositions. a) p ∧ ¬p b) p ∨ ¬p c) (p ∨ ¬q) → q d) (p ∨ q) → (p ∧ q) e) (p → q) ↔ (¬q → ¬p) f) (p → q) → (q → p)

Show that ¬(p ⊕ q) and p ↔ q are logically equivalent

Show that (p → q) ∧ (p → r) and p → (q ∧ r) are logically equivalent

Show that (p → q) ∧ (q → r) → (p → r) is a tautology

COUNTING METHODS mod4: 1. How many 5-digit number can be formed from digits 0 – 6 if: a. If repetition is allowed? b. If repetition is not allowed? c. If one (1) is not to be used as the 1st digit and repetition is not allowed? d…

PERMUTATIONS (mod4) 1. There are 6 people to be arranged in a line for a concert. How many arrangements are possible? 2. How many strings of length 5 can be formed using the letters QUALITY if a. Repetitions are not allowed? b. Re…

COMBINATIONS (mod4) 1. Patrick has assignments in 5 subjects. He can only do two assignments. In how many ways can he do two assignments? 2. In how many ways can a group of 5 men and 3 women be made out of a total of 10 men and 6 wo…

Solve the following. (mod4) 1. How many 3-digit number can be formed from digits 1 – 5 if: a. If repetition is allowed? b. If repetition is not allowed? 2. How many strings of length 4 can be formed using the letters COMPUTER if …

MATHEMATICAL INDUCTION AND RECURRENCE 5. If P(k) = k2 (k + 2)(k – 1) is true, then what is P (k + 1)? (2 pts) A. (k + 1)2 (k + 2)(k) B. (k + 1)2 (k + 2)(k) C. (k + 1)(k + 3)(k) D. (k + 1)2 (k + 3)(k) 6. Using the principle of m…

MATHEMATICAL INDUCTION AND RECURRENC Solve the following. (10 pts each) 1. Prove P(n) = n2 (n + 1) 2. Recurrence relation an = 2n with the initial term a1 = 2.

Let A = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4} and define a relation R on A as follows: For all x, y A, x R y ⇔ 3|(x − y). It is a fact that R is an equivalence relation on A. Use set-roster notation to write the equivalence classes…

f p is plotted versus a range of parameter value x the resulting defines ...a?

If the statement q ∧ r is true, determine all combinations of truth values for p and s such that the statement (q → [¬p ∨ s]) ∧ [¬s → r] is true.

Construct the disjunctive normal form of the proposition (p → q) ∧ (r ↔ p).

Construct the conjunctive normal form of the proposition (¬p ∧ q) ∨ (¬r ∨ ¬q).

-Show that ux=c1eax +c2e-ax solution of the difference equation ux+1-2uxcosha+ux-1

If the statement q ∧ r is true, determine all combinations of truth values for p and s such that the statement (q → [¬p ∨ s]) ∧ [¬s → r] is true. (Highlight your final answer not just the truth table)

An island has two types of inhabitants. Those of type A only ask questions whose correct answer is Yes. Those of Type B only ask questions whose correct answer is No. On the Island, you encounter Tom and Jerry who are inhabitants of t…

A survey was given to 140 students. On average, 2 out of 7 students floss every day. How many students do NOT floss every day?

There are 18 325 Computer and 415 English majors at colege . ( a ) How many ways to pick 5 math 125 computer and 300 English majors ? many ways to pick one Yepresenta tative who is either Math or computer . Eng / i * s * h ? ( 6 ) How

At a college of 600 students there are (n+5) enrolled in physics, (n+20) in Mathematics who play sports and (n-15) in chemistry and play sports. Create a Venn diagram to illustrate this information. Where n is your arid number

At a college of 600 students there are (n+5) enrolled in physics, (n+20) in Mathematics who play sports and (n-15) in chemistry and play sports. Create a Venn diagram to illustrate this information. Where n=8

At a college of 600 students there are (n+5) enrolled in physics, (n+20) in Mathematics who play sports and (n-15) in chemistry and play sports. Create a Venn diagram to illustrate this information. Where n is your arid number.

At a college of 600 students there are (n+5) enrolled in physics, (n+20) in Mathematics who play sports and (n-15) in chemistry and play sports. Create a Venn diagram to illustrate this information. Where n=8

If P(k) = k2 (k + 2)(k – 1) is true, then what is P (k + 1)?

. A special type of password consists of six(6) different letters of the alphabet, where each letter is used only once. How many different possible passwords are there? (3 pts) A. 244, 140, 625 B. 308, 915, 776 C. 165,765,600 D. …

Let p: priya is tall and q:priya is beautiful write the following statement in symbolic form. i) Priya is tall and beautiful ii) Priya is tall but beautiful iii) It is false that priya is short or beautiful

The relation 'is the father of' is________. a) reflexive b) irreflexive c) transitive d) symmetric

In a market, 50 women sell only Apple, 25 sell Apple and Banana and 50 sell Banana. Each woman sells at least one the two fruits. How many women are there?

A. COUNTING METHODS 1. How many strings of length 4 can be formed using the letters ABCDE if it starts with letters AC and repetition is not allowed? 2. There are 10 multiple choice questions in an examination. Each of the question…

We want to make a 6 character password such that the password must start and end with a digit. Moreover, one digit must be even and other must be odd. There must be one capital letter. How many such password can we make? Note: Charact…

Let n be 26. Construct a map of a continent having n different countries in such a way that four colours are needed to colour bordering countries in different colours. Using blue for the ocean (and possibly for some of the countries),…

Translate these system specifications into English where the predicate S(x, y) is “x is in state y” and where the domain for x and y consists of all systems and all possible states, respectively. a) ∃xS(x, open) b) ∀x(S(x, malfunc…

Express the negations of these propositions using quantifiers, and in English. a) Every student in this class likes mathematics. b) There is a student in this class who has never seen a computer. c) There is a student in this class…

Let W(x, y) mean that student x has visited website y, where the domain for x consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence. a) …

For each of these arguments, explain which rules of inference are used for each step. a) “Doug, a student in this class, knows how to write programs in JAVA. Everyone who knows how to write programs in JAVA can get a high-paying jo…

Identify the error or errors in this argument that supposedly shows that if ∃xP (x) ∧ ∃xQ(x) is true then ∃x(P (x) ∧ Q(x)) is true. a) ∃xP (x) ∨ ∃xQ(x) Premise b) ∃xP (x) Simplification from (1) c) P (c) Existential instantiation …

What is the Cartesian product A × B × C, where A is the set of all airlines and B and C are both set of allcities in USA? Give an example of how this Cartesian product can be used

A = Q, (a*b) = a + b

The set of all natural number whose square is more than 21.

A. COUNTING METHODS (5 pts each) 1. How many strings of length 4 can be formed using the letters ABCDE if it starts with letters AC and repetition is not allowed? 2. There are 10 multiple choice questions in an examination. Each of…

Determine whether the function f is a bijection from R to R ? Find fog and gof where f(x)=2x2+3 and g(x)=x=1

(A∪B) c (A∪B)c

Write notes on how to find cartesian products of sets and in you concluding page, site examples on the applications of set theory to solving real world business problems

(a) How many code words over a, b, c, d of length 20 contain exactly 10 a’s? (b) How many contain exactly 10 a’s and 5b’s.

4 What is the probability that an arrangement of a, b, c, e, f, g begins and ends with a vowel? 5 Helen wants to buy a bunch of flowers. There are five types of flowers available, namely roses, chrysanthemums, lilies, carnations and …

6 a) Show, using the pigeonhole principle, that in any set of 5 integers, at least two have the same remainder when divided by 4. (b) Use the extended pigeonhole principle to show that there are at least 3 ways of choosing 2 differen…

1 Let a relation R be represented by the following matrix MR = 1000001 1111000  1 0 1 1 0 1 0  MR=1 0 0 1 0 0 0  1 1 0 1 0 0 0  0 1 0 1 1 0 1 0111100   Determine whether R is (a) reflexive (b) irreflexive (c)…

Using Boolean algebra simplify the statement ¬(𝑟 → 𝑠) → (¬𝑟)

Use generating function to prove the identity n ∑ k=o r k s n−k = r +s n

Helen wants to buy a bunch of flowers. There are five types of flowers available, namely roses, chrysanthemums, lilies, carnations and tulips. How many possible bunches of 16 flowers can she choose if the bunch must contain no more th…

an = a 0.5n + n, a1 = 0, where n is a power of 2, is a linear recurrence relation. ( true / falsa ).

Consider the following relation R on A where A = {1, 2, 3, 4, 5} aRb ⇔ a b < min(a, b) For example, 2R4 since 2 4 = 1 2 and min(2, 4) = 2 and 1 2 < 2. (a) Draw the digraph of R (4) (b) Give a path of length 2 from 3, if…

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. {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}

Let S = {a1,a2,a3,...an}be a set of test scores. Prove using the the indirect method of proof that if the average of this set of test scores is greater than 90, then at least one of the scores is greater than 90.

Apply your knowledge gaining from this course in describing how you convert a function into a relation and a relation into a function?

COUNTING METHODS (15 pts) 1. How many 5-digit number can be formed from digits 0 – 6 if: a. If repetition is allowed? (3 pts) b. If repetition is not allowed? (3 pts) c. If one (1) is not to be used as the 1st digit and rep…

PERMUTATIONS (21 pts) 1. There are 6 people to be arranged in a line for a concert. How many arrangements are possible? (3 pts) 2. How many strings of length 5 can be formed using the letters QUALITY if a. Repetitions ar…

COMBINATIONS (24 pts) 1. Patrick has assignments in 5 subjects. He can only do two assignments. In how many ways can he do two assignments? (3 pts) 2. In how many ways can a group of 5 men and 3 women be made out of a total…

BINOMIAL COEFFICIENTS (20 pts) 1. Expand the (2𝑚 − 2𝑏)3 using binomial coefficient. (14 pts) 2. Find for the coefficient of a5b5; (a - 4b )10 (3 pts) 3. Find the 5th term after expanding the expression (3x – 4y)15 (3 pts) …

Solve the following. (35 pts) 1. How many 3-digit number can be formed from digits 1 – 5 if: a. If repetition is allowed? (3 pts) b. If repetition is not allowed? (3 pts) 2. How many strings of length 4 can be formed …

Determine the truth-values of the followings “ 1+1=4 if and only if earth is round” “ Milk is white if and only if birds lay eggs” “ Sky is white if and only if 1=0” “ 33 is divisible by 5 if and only if hours has four legs” “ …

translate to propositional logic “To get tenure as a professor, it is sufficient to be world-famous.”

Decide for each of the following relations whether or not it is an equivalence relation. Give full reasons if it is an equivalence relation, give the equivalence classes. A. Let a,b E Z. Define aRb if and only if (a/b)E Z. B. Let a …

The complete m-partite graph 𝑲𝒏𝟏,𝒏𝟐,……,𝒏𝒎 has vertices partitioned into m subsets of 𝒏𝟏, 𝒏𝟐, … … , 𝒏𝒎 elements each, and vertices are adjacent if and only if they are in different subsets in the partition. For example, if m=2, it is o…

Using mathematical induction prove the following: Σn k=1 k+4/k(k + 1)(k+2) = n(3n + 7)/2(n+1)(n+2)

Let A, B, and C be sets. Show that (B − A) ∪ (C − A) = (B ∪ C) – A by using Venn diagram?

If the statement ∧ r is true, determine all combinations of truth values for p and s such that the statement (q→[¬p∨s])∧[¬s→r] is true.

A∩(B−C)

Determine the truth value of the following statement, if the domain consists of all real numbers. Explain your answer in one sentence only. ∃x(x2=2)

Let a and b be two Natural Numbers, such that the greatest common divisor of a and b is 63, and the least common multiple of a and b is 44452800. If ’b’ is an odd number, what is the minimum value of ’a’ possible? [Hint: a · b = gcd(a, b) · lcm(a, b)]

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Question ID: mtid-5-stid-8-sqid-1387-qpid-1125