Solved: A number of students prepared for an examination in physics, chemistry and mathematics. Out of this number, 15 took physics, 20 took chemistryand 23 took mathematics, 9 students took both chemistry and mathematics, 6 student took both physics and mathematics and all those who took physics also took chemistry. One student fell ill and failed to write any of the papers. I. Represent the information on a Venn diagram II. How many students took all three subjects? III. How many students took exactly one of the subject? IV. How many students took exactly two of the subjects? V. How many students prepared for the examination?

12 Question 3: Mathematical proficiency and the construction of mathematics ideas. To answer this question, you need to complete the table below, and also study paragraphs 2.7-2.13 in the study guide: the most important is that yo…

Un+2-5Un+1-6Un=4^n+n

Define intersection of two Fuzzy set with example.

Mary Ann pays a monthly fee for her cell phone package which includes 700 minutes. She gets Billed an additional charge for every minute she uses the phone past the 700 minutes. During her first month, Mary Ann used 26 additional minu…

Let R be the relation on Z (the set of integers) defined by (x, y)  R iff x 2 + y2 = 2k for some integers k  0. Answer questions 13 to 15 by using the given relation R. Question 13 Which one of the following is an ordered p…

Let R be the relation on Z (the set of integers) defined by (x, y)  R iff x 2 + y2 = 2k for some integers k  0. Question 15 R is not antisymmetric. Which of the following ordered pairs can be used together in a counterexam…

Qu 01: Model the following situations as (possibly weighted, possibly directed) graphs. Draw each graph, and give the corresponding adjacency matrices. (i) Rokey and Jane are friends. Rokey is also friends with Kevin and Ferdy. Jane …

Prove that (ab)n=anbn(ab)n=anbn is true for every natural number n

Use induction to prove for all n∈Nn∈N that n∑k=02k=2n+1−1.

Prove that 7n−17n−1 is a multiple of 6 for all n∈N.

1.Find a possible statistical problem inside your home relating to a discrete random variable lesson, explain in a paragraph form…..(2-3 sentence will do) 2.Find the possible random variable 3.Create a probability distribution 4.Co…

Exhibit a cycle of length 9 in this graph

Calculate the number of vertices in a full 5-ary tree with 45 internal vertices.Also find out the number of leaves.

Simplify the Boolean Function using K Map (5 pts each) and draw the logic diagram (5 pts each) a. F(a, b, c) = ∑ (0, 1, 2, 4, 5, 7) b. F(a, b, c, d) = ∑ (0, 1, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14) c. F (w, x, y, z) = ∏ (1, 2, 7, 11, 1…

In a school 100 students have access to three software packages A, B, C. Where 28 didn’t use any software, 8 used only package C, 26 used only package A,7 used package B, 10 used all three packages, 13 used both A and C a) Draw a …

Question 1: (2 marks) (C1) Let’s consider a propositional language where A =“I study data structure”, B =“I study Bioinformatics”, C =“I study programming”, D =“I study discrete math” a. “I study Bioinformatics if I study progra…

1. What are the converse, inverse and contrapositive of the statement “If you try hard, then you will win” Converse Inverse Contrapositive b. Determine whether this proposition is a tautology, contradiction or contingency using t…

(Direct proof) A claim is given as a quantified statement: “The product of two odd numbers is an odd number” a) (1 point) Write the domain of the variables: b) (4 points) Write the statement using quantifiers and an implication of pro…

Find the simplest form for the following boolean expressions: 1) (A.B'.C')+(A'.B'.C')+(A'.B.C')+(A'.B'.C) 2) (A'.B.C)+(A'.B.C)+(A.B.C')+(A.B'.C')+(A'.B.C')+(A'.B'.C') 3) (A+B+C)(A+B'+C')(A+B+C')(A+B+C') ( ' = Not )

£(x - 50) =60,£(y-20) = -20,£(x -50)^2 = 450,£(y - 20)^2 = 200,£(x - 50)(y - 20) = -100

Find the simplest form for the following boolean expressions: 1) (A.B'.C')+(A'.B'.C')+(A'.B.C')+(A'.B'.C) 2) (A'.B.C)+(A'.B.C)+(A.B.C')+(A.B'.C')+(A'.B.C')+(A'.B'.C') 3) (A+B+C)(A+B'+C')(A+B+C')(A+B+C') ( ' = Not )

£(x - 50) =60,£(y-20) = -20,£(x -50)^2 = 450,£(y - 20)^2 = 200,£(x - 50)(y - 20) = -100

What are the truth sets of the predicates P(x), Q(x), and R(x), where the domain is the set of integers and 1. P(x) is “|x| = 4,” 2. Q(x) is “x2 = 16,” 3. R(x) is “|x| = x”

State which of the following are not a function from R to R and why. (a) f(x) = 1/x (b) f(x) = 1/(1+x) (c) f(x) = (x)½ (d) f(x) = ±(x2+1)½ (e) f(x) = sin(x) (f) f(x) = ex

Find out which of the following functions from R to R are (i) One-to-one, (ii) Onto, (iii) One-to-one correspondence. (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = |x| (c) f: R—>R defined by f(x) = x + 1 (d) f…

Draw graphs of the following functions. (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = |x| (c) f: R—>R defined by f(x) = x + 1 (d) f: R—>R defined by f(x) = – 3x+4 (e) f: R—>R defined by f(x) = Floor(x) (f) …

What will be the inverse of the following functions from R to R? (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = x + 1 (c) f: R—>R defined by f(x) = – 3x+4 (d) f: R—>R defined by f(x) = x^3 (e) f: R—>R defined …

What will be the composition of two functions f ° g from R to R, (a) f(x) = 2x + 3, g(x) = 3x + 2 (b) f(x) = 2x + 3, g(x) = sin (x) (c) f(x) = sin (x), g(x) = 2x + 3 (d) f(x) = sin (x), g(x) = x2 (e) f(x) = |x|, g(x) = 2x + 3

Show that the following relations are Equivalence relations. Find out the Equivalence classes. (a) ρ = {(a, b) | a – b is an integer} on the set of real numbers R (b) R = {(a, b) | a = b or a = –b } on the set of integers Z (c) Con…

Show that the following relations are Partial order relations. (a) R on the set of integers Z, defined by, R = {(a, b) | a ≤ b} (b) R on the set of integers Z, defined by, R = {(a, b) | a ≥ b} (c) R on the set of positive integers …

Question 1. [20 Marks] a) Let p, q, and r be the propositions: p = "the flag is set" q = "I = 0" r = "subroutine S is completed" Translate each of the following propositions into symbols, using the l…

Every function is a relation, but the converse is not true.”--True or false? Justify with an example.

Show that a fuzzy relation R on a set U is anti-symmetric if and only if each of its alpha-cuts is an anti-symmettric relation on U

a) Write each statement in symbolic form using p, q and r. If I study, then I will not fail mathematics. If I do not play basketball, then I will study. But I failed mathematics. Then, test the validity of the fol…

Write down the relation R on the Power-set of S = {1, 2, 3, 4}, defined by, R = {(A, B) | A and B are subsets of S and they have the same cardinality} using any one of the methods: (i) set of ordered pairs, (ii) directed graphs, (ii…

(I) A relation R on a set S is called irreflexive, if no element in S is related to itself, that is if for every a in S, (a, a) is not in R. Which of the relations in Q. No. 5 is irreflexive?

A relation R on a set S is called asymmetric if (a, b) is in R implies that (a, b) is not in R. Which of the relations in Q. No. 5 is asymmetric?

Let R = {(0, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 2), (3, 0)} be a relation on the set {0, 1, 2, 3}. Find the (a) Reflexive closure of R (b) Symmetric closure of R (c) Transitive closure of R

Show that the following relations are Equivalence relations. Find out the Equivalence classes. (a) ρ = {(a, b) | a – b is an integer} on the set of real numbers R (b) R = {(a, b) | a = b or a = –b } on the set of integers Z (c) Con…

Show that the following relations are Partial order relations. (a) R on the set of integers Z, defined by, R = {(a, b) | a ≤ b} (b) R on the set of integers Z, defined by, R = {(a, b) | a ≥ b} (c) R on the set of positive integers …

Question 1. [20 Marks] a) Let p, q, and r be the propositions: p = "the flag is set" q = "I = 0" r = "subroutine S is completed" Translate each of the following propositions into symbols, using the l…

Every function is a relation, but the converse is not true.”--True or false? Justify with an example.

Show that a fuzzy relation R on a set U is anti-symmetric if and only if each of its alpha-cuts is an anti-symmettric relation on U

a) Write each statement in symbolic form using p, q and r. If I study, then I will not fail mathematics. If I do not play basketball, then I will study. But I failed mathematics. Then, test the validity of the fol…

A fair six-sided dice is thrown and the scores are noted. Event X: The total of the two scores is 4. Even Y: The first score is 2 or 5. a) Construct the table by showing the sample spaces.

a) Freddie has 6 toys cars and 3 toy buses, all different. i) Freddie arranges these 9 toys in a line. Find the number of possible arrangements if the buses are all next to each other.

a) Freddie has 6 toys cars and 3 toy buses, all different. i) Freddie arranges these 9 toys in a line. Find the number of possible arrangements · if there is a car at each end of the line and no bus…

What is 100010002 - 00011011002

Suppose today is thursday 23 April 2020 use modular arithmetic to determine with a correct reason which day of the week will it be on 23 April 2036

Suppose U = {1, 2, 3, 4, 5, a, b, c} is a universal set with the subset A = {a, b, c, 1, 2, 3, 4}. Answer questions 1 and 2 by using the given sets U and A. Question 1 Which one of the following relations on A is NOT functional? …

Question 3 Let G and L be relations on A = {1, 2, 3, 4} with G = {(1, 2), (2, 3), (4, 3)} and L = {(2, 2), (1, 3), (3, 4)}. Which one of the following alternatives represents the relation L ￮ G = G; L? 1. {(2, 3), (3, 3)} 2. {(1,…

Answer questions 4 to 7 by using the given functions g and f. Hint: Drawing graphs of f and g before answering the questions, may assist you. Keep in mind that g  Z+  Q and f  Z+  Z+. Please note that graphs will not be asked …

Let A = {□, ◊, ☼, ⌂} and let # be a binary operation from A  A to A presented by the following table: # □ ◊ ☼ ⌂ □ □ ◊ ☼ ⌂ ◊ ◊ □ ◊ □ ☼ ☼ ◊ ☼ ⌂ ⌂ ⌂ □ ⌂ ⌂ Answer questions 10 and 11 by referring to the table for #. Question …

Question 14 Consider the truth table for the connective ‘’ with two simple declarative statements p and q. p q p  q T T T T F F F T F F F T (a) Which one of the given alternatives represents ‘’ as a binary operation on …

Question 15 Let p, q and r be simple declarative statements. Which alternative provides the truth values for the biconditional ‘’ of the compound statement provided in the given table? Hint: Determine the truth values of p → r, q…

Question 16 Consider the following quantified statement: ∀x ∈ Z [(x2 ≥ 0) ∨ (x2 + 2x – 8＞0)]. Which one of the alternatives provides a true statement regarding the given statement or its negation? 1. The negation ∃x ∈ Z [(x2 …

Question 17 Consider the following proposition: For any predicates P(x) and Q(x) over a domain D, the negation of the statement ∃x ∈ D, P(x) ∧ Q(x) is the statement ∀x ∈ D, P(x) → ¬Q(x). We can use this truth to write the nega…

Question 22 Consider the following statement: ∀x  Z, [(2x + 4 > 0)  (4 - x 2 ≤ 0)] The negation of the above statement is: ¬[∀x  Z, [(2x + 4 > 0)  (4 - x 2 ≤ 0)]] ≡ ∃x  Z, ¬[(2x + 4 > 0)  (4 - x 2 ≤ 0)] ≡ ∃x  Z, [¬(2x …

Question 24 Consider the statement If n is a multiple of 3, then 2n + 2 is not a multiple of 3. The converse of the given statement is: If n is not a multiple of 3, then 2n + 2 is a multiple of 3. 1. True 2. False Question 25…

Draw graphs of the following functions. (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = |x| (c) f: R—>R defined by f(x) = x + 1 (d) f: R—>R defined by f(x) = – 3x+4 (e) f: R—>R defined by f(x) = Floor(x) (f) …

Find out which of the following functions from R to R are (i) One-to-one, (ii) Onto, (iii) One-to-one correspondence. (a) f: R—>R defined by f(x) = x (b) f: R—>R defined by f(x) = |x| (c) f: R—>R defined by f(x) = x + 1 (d) f…

What are the differences between relations and functions?

Let A be {a, b, c}. Let the relation R be {(c, b), (a, a), (b, c)}. Which of the following statements about R is true? a. R is not reflexive, is not symmetric, and is not transitive. b. R is reflexive, is symmetric, and is not transi…

{(0,-5),(1,-4),(2,-3),(3,-2),(4,-1),(5,0)} Is this a relation

You have been working as a mathematical analyst for the Everest Statistics Bureau Pvt. Ltd. operating in Kathmandu Nepal. The Bureau helps the Government of Nepal by providing statistics in developing Public Policy Planning, Collectin…

a) (p ↔ q) ⊕ (¬p ↔¬r) b) (p → q) ∧ (¬p → r)

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

Prove or disprove ¬(¬p -> q) =¬(p Ú q) is correct.

Is (p>q)>[(p>q)>q] a tautology? Why or why not?

25. Solve recurrence relation an+3=3an+2+4an+1-12an for n20 with a0-0,al--11,a2--15

i) Which type of relation is shown in below expression R1 = { (a,b) | a = b }

Find each of the function below, indicate whether the function in onto, on-to-one neither or both. If the function is not onto or nor one-to-one, give an example showing why …

Find each of the function below, indicate whether the function in onto, on-to-one neither or both. If the function is not onto or nor one-to-one, give an example showing why . H;Z Z. h(x…

An examination paper consists of 5 questions in section A and 5 questions in section B. A total of 8 questions must be answered. In how many ways can a student select the questions if he is to answer at least 4 questions from section …

Prove the following statement by induction. For all nonnegative integers nn, 3 divides n^3 +2n +3n. State the mathematical induction and show your work clearly.

Given R = (0^*10^+)^+1^*(0∗10+)+1∗ and S =(1^*01^+)^*(1∗01+)∗ . a) Give an example of a string that is neither in the language of R nor in S. [2marks] b) Give an example of a string that is in the language of S but not …

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.

Find each of the function below, indicate whether the function in onto, on-to-one neither or both. If the function is not onto or nor one-to-one, give an example showing why G;R R. g(x)=x^3

i) Which type of relation is shown in below expression R1 = { (a,b) | a = b }

Show that ¬ (P Q) (P V Q) Λ ¬(P Λ Q) (P Λ ¬Q) V (¬ P Λ Q) without using truth table

You visit an island where three triplet brothers named Lanister, Lewis and Tom, live. They are indistinguishable in appearance, but Lanister and Lewis, are knaves whereas Tom is a knight. One day you meet one of the three on the stree…

Q1. Find the truth tables for the following statement forms: 1. p∨ ∼ q 2. p ∨ (q ∧ r ) 3. (p ∨ q) ∧ (p ∨ r )

Determine if the following system specifications are consistent or not? “You get good food if and only if you work attentively in the kitchen. Having no Sui gas is a sufficient condition for you to not get good food. If the weather …

Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contraposition. b) a proof by contradiction

Let R be the relation on Z (the set of integers) defined by (x, y)  R iff x 2 + y2 = 2k for some integers k  0. Answer questions 13 to 15 by using the given relation R. Question 13 Which one of the following is an ordered p…

A committee of 3 individuals decides on issues for an organization. Each individual votes either a YES or a NO for each proposal to pass. A proposal is passed if it receives at least two YES votes . Design a circuit that determines if…

How many elements does A=0 have ?

Suppose U = {1, 2, 3, 4, 5, a, b, c} is a universal set with the subset A = {a, b, c, 1, 2, 3, 4}. Answer questions 1 and 2 by using the given sets U and A. Question 1 Which one of the following relations on A is NOT functional? 1…

. Check the validity of the following arguments (a) Hayder works hard. If Hayder works hard, then he is a dull boy. If Hayder is a dull boy, then he will not get the job. therefore, Randy will not get the job. (b) If it does not rain …

Part 4 1. Consider the K-Maps given below. For each K- Map i. Write the appropriate standard form (SOP/POS) of Boolean expression. ii. Design the circuit using AND, NOT and OR gates. iii. Design the circuit only by using • NAND gates …

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Question ID: mtid-5-stid-8-sqid-2760-qpid-1293