Solved: (a) In how many ways can a committee of 3 faculty members and two students be selected from 7 faculty members and 8 students (b) How many ways are there to distribute 12 different books among 15 people if no person is to receive more than one book

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…

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 …

Using laws of logic solve the following compound propositions. Also indicate the names of laws. [𝑝 ∧ (¬𝑝 ∨ 𝑞)] ∨ [𝑞 ∧ ¬(𝑝 ∧ 𝑞)]

The bits string for A abd B are 1100101001 and 1011010110 respectively. Use bit string to find A intersection B compliment

Find the generating function of recurrence relation an+1_an=3n ,n less than 0 where ao=1

We consider the statement <> then we can deduce that: A. If the number 13 is not raffled then Bruno becomes poor B. Since Bruno never became rich that means the number 13 has not been…

We consider the statement <> then we can deduce that: A. If the number 13 is not raffled then Bruno becomes poor B. Since Bruno never became rich that means the number 13 has not …

Let A be the set of natural numbers multiples of 20 and B the set of natural numbers multiples of 15. Which is the set A ∩ B? A.natural numbers multiples of 60 B. natural numbers multiples of 5 C. natural numbers multiples of 300 …

Write Inverse, Converse and Contrapositive, also apply implication law on the following statements: a)If it snows today. I wi…

: Let p and q be the propositions. p: “I bought a lottery ticket.” q: “I won the million dollar jackpot on Friday.” Express each of these propositions as an English sent…

Question 5: By using the rules of logical equivalences, show the propositions are logically equivalent: a) Determine whether (p → (q → r)) → (p ˄ q) → r) is Tautology. b) (p ∧ q) ∧ [(q ∧ ¬r) ∨ (p ∧ r)…

Q.No.5. [2+1+3] a) Draw a tree with n vertices with n+1 vertices of degree 2, n+2 vertices of degree 3, and n+3 vertices of degree 1. Where n is even digit of your arid number e.g 19-arid-234 take n=2

In a school, n+100 students have access to three software packages A, B andC n-1 did not use any software , n-2 used only packagesA n-3 used only packages B , n-4 used only packages…

Draw a tree with n vertices with n+1 vertices of degree 2, n+2 vertices of degree 3, and n+3 vertices of degree 1. Where n is 863

In a school, 863+100 students have access to three software packages A, B and C. 862 did not use any software. 861 used only packages A. 860 used only packages B. 859 used only packages C. 858 used all three pa…

a) Draw a tree with n vertices with n+1 vertices of degree 2, n+2 vertices of degree 3, and n+3 vertices of degree 1. Where n is 7

Make your own example with at least four restrictions and one conclusion. Description: p∨q∨r∨s→t=(p→t)∧(q→t)∧(r→t)∧(s→t)?

For any natural number n, prove the validity of given series by mathematical induction: 2(√(n+1)-1)<1+(1/√2)+⋯..+(1/√n)<2√n?

a) Draw a tree with n vertices with n+1 vertices of degree 2, n+2 vertices of degree 3, and n+3 vertices of degree 1. Where n is even digit of your arid number e.g 19-arid-234 take n=2

a) Draw a tree with n vertices with n+1 vertices of degree 2, n+2 vertices of degree 3, and n+3 vertices of degree 1. Where n is 158

In a school, n+100 students have access to three software packages A, B and C.n-1 did not use any software , n-2 used only packages A n-3 used only packages B , n-4 used only packages C n-5 used all three packages,n-6 …

a) Draw a tree with n vertices with n+1 vertices of degree 2, n+2 vertices of degree 3, and n+3 vertices of degree 1. Where n is even digit of your arid number e.g 19-arid-234 take n=2

(b) For any natural number n, prove the validity of given series by mathematical induction:

In a school 844 students have access to three software packages A, B, C. Where 743 didn’t use any software, 740 used only package C, 742 used only package A,741 used package B, 739 used all three packages, 738 used both A and C …

Produce a truth table for given Boolean expression (A+B'+C)(A+B+C)(A'+B+C')

Let R and S be relations on X. Determine whether each statement is true or false. If the statement is true, prove it; otherwise, give a counterexample. 1. If R is transitive, then R−1 is transitive. 2. If R and S are refl…

1. Three departmental committees have 6, 12, and 9 members with no overlapping membership. In how many ways can these committees send one member to meet with the president? 2. Two dice are rolled, one blue and one red. How ma…

Basic Counting Principle 6. How many different car license plates can be constructed if the licenses contain three letters followed by two digits if: a.) Repetitions are allowed; b.) repetitions are not allowed. 7. Two d…

Find the generating function of recurrence relation an+1_an=3n ,n less than 0 where ao=1

There are 18 mathematics majors and 325 computer science majors at a college. a) In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major?

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

Design a single error correcting code for m=3 & n=7

(a) In how many ways can a committee of 3 faculty members and two students be selected from 7 faculty members and 8 students (b) How many ways are there to distribute 12 different books among 15 people if no person is to receive more…

(a) Show that a simple connected graph with 7 vertices each of degree 4 is non-planar (b) Find χ(Kn) & χ(Cn)

Let g be a function from Z+ (the set of positive integers) to Q (the set of rational numbers) defined by (x, y) element of g iff y = (g is a subset of Z+ mapped with Q) and let f be a function on Z + defined by (x, y) element of f iff…

Let A = {1,2,3,4,6,8,9,12,18,24} be a non-empty set and R be the partial order relation of divisibility defined on A, i.e., If (a,b) ε A, then a divides b i. Draw the Hasse diagram of R. ii. Find the Maximal and Minimal elements in…

Use breadth-first search to produce a spanning tree for each of the simple graphs in Exercises 1 3-15. Choose a as the root of each spanning tree.

an-3an-1-4an-2=4.3n

(a) In how many ways can a committee of 3 faculty members and two students be selected from 7 faculty members and 8 students (b) How many ways are there to distribute 12 different books among 15 people if no person is to receive more than one book

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Question ID: mtid-5-stid-8-sqid-2773-qpid-1330