Solved: In general, a statement involving n variables x1 , x 2 ,…, x n can be denoted by P( x1 , x 2 ,…, x n ) and is the value of the predicate P at the n-tuple (x1 , x2 ,…, x n ) . (a) Suppose Q(x , y) denotes the statement “ x=y+3 .” What are the truth values of the propositions Q(1,2) and Q(3,0)? (b) Let R(x , y ,

Let A = {1, 2, 3, 4} and let R be a relation on A such that R = {(1, 1),(2, 2),(3, 3),(4, 4),(1, 2),(2, 3),(3, 2),(2, 1)} Is R transitive? Symmetric? Reflexive?

Let A = {1, 2, 3, 4} and let R be a relation on A such that R = {(1, 1),(2, 2),(3, 3),(4, 4),(1, 2),(2, 3),(1, 3)} Is R transitive? Symmetric? Reflexive?

Let C = {1, 4, 5} and D = {2, 7}. • List the elements of C × D. • If there are 140 elements in D × A, how many elements are in A? • Give an example of a relation from D to C that is not a function. Draw a graph of the relation. Expla…

Prove by mathematical induction that: Where "E" is the summation icon. n i E E j = 1/6n(n+1)(n+2) i = 1 j=1

Which of the following statements are true? Give reasons for your answers. i) Set {p,q,r} and {p,r, q, p} are equal.

Use mathematical induction to prove that 1^3 + 2^3 + ... + n^3 = =(n(n+1)/2)^2 for all integers n ≥ 1

Let U be the set of positive integers 1, 2, 3, ... etc., A be the set of odd positive integers and B be the set of even positive integers. Verify De Morgan's laws.

A school bus is transporting athletes home from an athletics meeting.of these athletes 15% took part in hurdles,20% took part in the long jump event ,35% took part in the relay races,30% ran 100- m sprint and 25% took part in high jum…

Write the converse and the contrapositive of the following conditional: Conditional: If it is sunny then it is daylight. Converse: Contrapositive:

A school bus is transporting athletes home from an athletics meeting. Of these athletes 15% took part in hurdles,20% took part in the long jump event 35%,took part in relay races,30% ran the 100-m sprint and 25% took part in the high-…

The contrapositive of 'there exist y€z such that p(y) is true' is ' there exist x€z such that p(x) is true'. Is the statement true or false? Justify your answer.

1. Expand in a two-element universe (the elements are named 'a' and 'b') (a) ~(x) ((Fx v Gy) v Ka) (b) (x) ~ (Kx v Ka) (c) (Ex) (Cy v (Fx --> ~Ga)) 2. For the following wffs, indicate which variables are free and which are bound (…

draw the graphs that have the following vertices and edges V = {a, b, c}, and E = {{a, b}, {b, c}}.

draw a graph that has the following vertices and edges. V = {x1, x2, x3, x4} and E = {{x1, x3, }, {x1, x4}, {x2, x4}}

How many 20 digit binary numbers have four 1's in them?

Show that the following argument form is valid. p --> q q --> r ∴ p --> r

a) A market researcher investigating consumers preference for three brands of beverages namely; coffee, tea and cocoa in Thika town gathered the following information. From a sample of 800 consumers, 230 took coffee, 245 took tea and …

What is the value of k after the following code has been executed? k := 0 for i1 := 1 to n1 for i2 := 1 to n2 . . . for im := 1 to nm k := k + 1

According to information obtained from mathematics department regarding three mathematics units done by 100 students, those who are doing calculus are 45, those doing discrete are 49 and those doing statistics are 38. Those doing calc…

Give that the universal set • µ = (1,2,3,4,5,6,7,8,9,10) , p = (1,2,4,6,10) and Q = (2,3,6,9). Show that (P U Q)’= P’ n Q’

“If it is a wild animal, then it is dangerous. If it is dangerous, then it will hurt you. However, it is not dangerous. Therefore, it is not a wild animal. Give a step-by-step argument using valid Rules of Inference

Prove the following If a is odd and b is even, then a2 – b2 is an odd number

P:the cow is old Q:it is dying Express each of these prepositions as an English sentence p v q. p ^q. p v(~q)

If the domain is the set of words {violet, indigo, blue, green, yellow, orange, red}, which of the following sentences is/are true? (P)∀x(if x does not contain the letter ‘e’, then x contains the letter ‘n’) (Q) ∃x(if x contains the…

1. Let x = (1,2,3,4), Y = (2,3,5) and Z =(4,5,6). verify the following: a) x U y = y U x. b) (x U y) U z = x U (y U z).

How many 10 digit binary numbers have four 1's in them?

Go¨del′s completeness theorem asserts that --- The first order proof system with Peano's axioms proves every statement true in the standard model Peano's axioms form a consistent set of formulae The first order proof system can pro…

How to go back in a table so the common difference is adding each time by 5,7,9 etc

In a game of chess, a queen can travel any number of squares in a straight line- horizontally, vertically or diagonally. Moving the queen from queen (q) to king (k) visiting each square exactly once with the minimum number of moves po…

3.The sets (A-B), \\(A\\cap B\\) and (B-A) are mutually disjoint implies… a.the difference of any two is the null set b.the intersection of any two is the null set c.the union of any two is the null set d…

7.The family of all the subsets of any set S is called a.the power set of S b.the null set of S c.the identity set of S d.the cardinality set of S 8.Let E = {2, 4, 6, ...}. What is the compliment of the set E? a.odd numbers b.prime n…

1. For integers a and b,if ab is odd, then a and b are odd. 2. If xy=(x +y)^2 / 4 ,then x=y. prove the following 2 statements, state the method used and explain all necessary steps. 3.Suppose that factorial is the Python function de…

Prove or disprove: If A, B, and C are nonempty sets, and A×B = A×C, then B = C.

If the domain of discourse is all integers, find a counterexample*, if possible, to the following universally quantified statements: a. ∀x∃y(x = 1/y) b. ∀x∃y(y2 −x < 100) c. ∀x∀y(x2= y3)

Suppose g : A → B and f : B → C are functions. a. Show that if f ◦g is onto, then f must also be onto. b. Show that if f ◦g is one-to-one, then g must also be one-to-one. c. Show that if f ◦g is a bijection, then g is onto if and only…

Write in expression in p, q, and logical connectives which gives the following truth table: p q ? p= T T T F, q= F T F F, s= F T F F

Express the negation of the following statements WITHOUT using the negation symbol: a. ∀x(−2 < x <3) b. ∀x(0 ≤ x <5) c. ∃x(−4 ≤ x ≤1) d. ∃x(−5 < x <−1) e. ∀x∃y(x2 < y)

Use the principle of induction Prove that 2 1 n 2 n 2 n n 1 > + ∀ > − ,

Show that 1n^3+2n+3n^2 is divided by 2 and 3 for all positive integers n

which of the following statement is true a. \\((p\\wedge q)\\vee (p\\vee r)=p\\vee (q\\wedge r)\\) b. \\((p\\vee q)\\wedge (p\&b…

What reads "the goods are are standard if and only if the goods are expensive"?

Let A = {2, 3, 4} and B = {6, 8, 10} and define a relation R from A to B as follows: For all (x, y)∈ A ×B, (x, y)∈ R means that is an integer. a. Is 4 R 6? Is 4 R 8? Is (3, 8) ∈R? Is (2, 10) ∈R?

At the Keep in Shape Club, 35 people swim, 24 play tennis, and 27 jog. Of these people, 12 swim and play tennis, 19 play tennis and jog, and 13 jog and swim. Nine people do all three activities. How many members are there altogether?

Construct a directed graph for the board members of the company if the President can influence the Director of research and Development, the Director of Marketing, and the Director of Operations; the Director of Research and Developme…

The famous detective TVTHREE, Kara Singh Walla was called in to solve a baffling murder mystery. He determined the following facts: A. Alan, the murdered man, was killed by a blow on the head with brass candlestick. B. Either Alan’s…

Consider the sequences (rn) and (sn) defined recursively by r0 = 1, s0 = 0, and rn+1 = rn/2, sn+1 = sn + rn+1 for n ≥ 0. (a) What are the formulas for the nth terms rn and sn of these sequences? (b) What is the floating point binary r…

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

At the beginning of the first day (day 1) after grape harvesting is completed, a grape grower has 8000 kg of grapes in storage. At the end of day n, for n = 1, 2, . . . , the grape grower sells 250n/(n + 1) kg of their stored grapes a…

how many 4-letter words with or without meaning ,can be formed out of the letters of the word, "LOGARITHMS" ,if repetition is not allowed?

Let f : A → B and let X,Y be subsets of the domain A. For any Z ⊆ A, define the image of Z under f to be the set f[Z] = {b ∈ B|∃z ∈ Z(f(z) = b)}. a. Show that f[X ∪Y] = f[X]∪f[Y]. b. Give an example of a function f and subsets X,Y o…

Give a direct proof, as well as a proof by contradiction, of the following statement: ‘ B A ∩ B ⊆ A ∪ for any two sets A and B

5.Let f be a function of A into B. If every member of B appears as the image of at least one element of A, then we say the function f is a.surjective functions b.constant function c.injective functions d.identity functions 6.Let f be…

1.In class of 40 students, 38 offer maths, 24 offer English. Each of the students offer at least one of the two subjects. How many students offer both subjects? 2. If A={ 3 ,4 ,5 ,9 and B={1,2,6,9,7}. Find (I) A–B (ii)B n A.

Let p and q bethepropositions“Theelectionisdecided” and“Thevoteshavebeencounted,”respectively.Express each of these compound propositions as an English sentence

Calculate (by hand) the appropriate truth table to prove or disprove the following: (p → q) ∧ (¬p → r) ⇒ (q ∨ r) If it is invalid, give a counterexample; otherwise, explain why it is valid.

(a) Use rules of inference and laws of logical equivalence to prove the following: (p → q) ∧ (r → s) ∧ [t → ¬(q ∨ s)] ∧ t ⇒ (¬p ∧ ¬r) (b) Use the MATLAB program truth.m and a modified version of propos.m to verify the valid- ity of th…

Use mathematical induction to prove that 4 is a factor of 9n − 5 n for all integers n ≥ 1.

Decide whether or not the following quantified propositions are true or false, where in each case the universal set is the set of positive integers N+. Justify each of your conclusions with a proof or a counterexample. (a) ∀n(2 n ≥ n …

(a) In how many different ways can the letters of the word wombat be arranged? (b) In how many different ways can the letters of the word wombat be arranged if the letters wo must remain together (in this order)? (c) How many differen…

In an exam, a student is required to answer 10 out of 13 questions. Find the number of possible choices if the student must answer: (a) the first two questions; (b) the first or second question, but not both; (c) exactly 3 out of the …

Police report that 90% of drivers stopped on suspicion of drunk driving are given a breath test, 11% are given a blood test, and 8% are given both. (a) In this context, define two events A and B. (b) Write the given information in pro…

Use properties of Boolean algebra to simplify the following Boolean expression (show- ing all the steps): (x + y 0 )(x 0 + y 0 ) 0

(a) In how many different ways can the letters of the word wombat be arranged? (b) In how many different ways can the letters of the word wombat be arranged if the letters wo must remain together (in this order)? (c) How many differen…

In an exam, a student is required to answer 10 out of 13 questions. Find the number of possible choices if the student must answer: (a) the first two questions; (b) the first or second question, but not both; (c) exactly 3 out of the …

Use properties of Boolean algebra to simplify the following Boolean expression (show- ing all the steps): (x + y') (x + y')'

Xn i=1 i2 i = (n − 1)2n+1 + 2

Use strong induction to prove that every integer n > 5 can be written as n = 4x + 3y for non-negative x, y ∈ Z.

There are 60 science student in a secondary school 35 of whom study chemistry and 30 study technical drawing.12 out of those students study biology and chemistry but not technical drawing.10 study chemistry but neither biology nor tec…

In a class, 20 students study agric, 21 study biology and those who study chemistry are 5 more than those who study agric. it is also known that 3 students study agric only and 4 students study only chemistry. Half as many students wh…

Find the total number of ways in which 10 different cakes can be divided into two parcels each containing the same number ?

A number of students prepared for an examination in physics, chemistry and mathematics. Out of this number, 15 took physics, 20 took chemistry and 23 took mathematics. 6 took both physics and mathematics and all those who took physics…

Use natural deduction to derive the conclusion in each problem. Use conditional proof or indirect proof as needed: 1. (x)(Dx ⊃ x = a) 2. (x)[Ex ⊃(Dx • x = e)] / (∃x)(Ex ⊃ a = e)

In how many ways can the word DISCRETE be arranged?

Given the a Boolean function F(x,y,z)=(x+ y').z, write the sum-of-products expansion of F where all the variables x, y and z are used. Hint: Use some Huffington postulates, Boolean theorems and the unit property: p+p'=1

give an example of a choice function on the collection. X = {{0.1},{2,3},{4,5}}

how many choice function we could we define on the set X X= {{0,1},{2,3},{4,5}}

Use mathematical induction to prove that ∀n∈N P(n):1·2·3+2·3·4+···+n(n+1)(n+2)=n(n+1)(n+2)(n+3)/4

given 2 sets A and B, use membership table to show that (A-B)∪(B-A)=(A∪B) -(A∩B)

Develop truth tables and its corresponding Boolean equation for the following scenarios. i. ''If the driver is present AND the driver has NOT buckled up AND the ignition switch is on, then the warning light should turn ON.'' ii. If it…

1. Develop truth tables and its corresponding Boolean equation for the following scenarios. i. ''If the driver is present AND the driver has NOT buckled up AND the ignition switch is on, then the warning light should turn ON.'' ii. If…

Give partimos of R having (a) one block, (b) two blocks, (c) three blocks, (d) infinity many blocks

List all elements of the following sets as a set. All answers must be exact and not rounded 1.{q is an integer | q is a factor of 231} Describe the following sets using proper set-builder notation as explained in your book. You may n…

2. Produce truth tables for given Boolean expressions. i. A̅ B̅ C+A B̅ C̅ + ABC +A̅ B C̅ ii. (A+B̅+C)(A+B+C)(A̅+B+C̅)

Use the symbols ~, ^ and v, and write the following statements. both p or q and r

For the language L = {acnb | n > 0 }, which one of the following strings is not a part of the language ?

Part 2 1. Write the multisets of prime factors for the given numbers. I. 160 II. 120 III. 250 2. Write the multiplicities of each element of multisets in part 2(1-I, ii,iii) separately. 3. Find the cardinalities of each multiset in pa…

1. Describe the characteristics of different binary operations that are performed on the same set. 2. Justify whether the given operations on relevant sets are binary operations or not. i. Multiplication and Division on se of Natural …

Build up the operation tables for group G with orders 1, 2, 3 and 4 using the elements a, b, c, and e as the identity element in an appropriate way. 2. i. State the Lagrange’s theorem of group theory. ii. For a subgroup H of a group G…

1. Check whether the set is a group under the binary operation ‘*’defined as for any two elements . 2. i. State the relation between the order of a group and the number of binary operations that can be defined on that set. ii. How man…

1. Develop truth tables and its corresponding Boolean equation for the following scenarios. i. ''If the driver is present AND the driver has NOT buckled up AND the ignition switch is on, then the warning light should turn ON.'' ii. If…

1. Discuss two real world binary problems in two different fields using applications of Boolean Algebra

Discuss two examples on binary trees both quantitatively and qualitatively.

2. Function converts Fahrenheit temperatures into Celsius. What is the function for opposite conversion?

1. State the Dijkstra’s algorithm for a directed weighted graph with all non-negative edge weights.

1. Construct a proof for the five color theorem for every planar graph. 2. Discuss how efficiently Graph Theory can be used in a route planning project for a vacation trip from Colombo to Trincomalee by considering most of the practic…

If A= (a,b,c) then generate P(A) and 3 partition sets for A.

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