Solved: A pair of dice is loaded. The probability that a 4 appears on the first die is 2/7, and the probability that a 3 appears on the second die is 2/7. Other outcomes for each die appear with probability 1/7. What is the probability of 7 appearing as the sum of the numbers when the two dice are rolled?

Describe two situations that can be modelled with a Graph. Describe the graph model that you will use. Describe two situations that can be modelled with a Tree. Describe the tree model that you will use.

Let R = {(2, 2), (2, 4), (3, 3), (3, 6), (3, 12), (4,2), (6, 3)} where if a is a factor b} (i) Draw the digraph for the relation P. (ii) Find the domain and range of the relation.

Q:1. Determine whether each set is a function from X = {1,2,3,4} to Y = {a,b,c,d}. If it is a function, find its domain and range, draw its arrow diagram, and determine if it is one-to-one, onto or both. a) {(1,a,),(2,a),(3,c),(4,b)} …

Q : 1) Determine whether each function is one-to-one. The domain of each function is the set of all real numbers. If the function is not one-to-one, prove it. Also, determine whether f is onto the set of all real numbers. If f is not …

Q : 5). If f is function from A to B and g is function B to C and both f and g are onto. Show that go f is also onto. Is go f one-to-one if both f and g are one-to-one. Q : 6) Let f, g and h: R → R be defined by (R is the set of rea…

Construct the call graph for a set of seven telephone numbers 555-0011, 555-1221, 555-1333, 555-8888, 555-2222, 555-0091, and 555-1200 if there were three calls from 555-0011 to 555-8888 and two calls from 555-8888 to 555-0011, two ca…

Discrete Mathematics - Languages and Machines Need to solve Finite State Automaton i.e, Your task is to design a binary finite state automaton (FSA) to accept all strings that represent valid messages (for your particular codes and pa…

For any two sets A and B , in a universal set U , prove that A ⊆ B ⇔ A ∪ B = B.

a) Simplify the Boolean function F = AB + (AC)′ + AB ′C(AB + C).

Implement the 3-variable function F (A,B,C) = (0,2,4,7) with a multiplexer

a. Minimize the following problems using the Karnaugh maps method. Z = f (A, B, C) = + B + AB + AC

Using the Karnaugh map method obtain the minimal sum of the products and product of sums expressions for the function F (A, B, C, D) = Σ (1, 5, 6, 7, 11, 12, 13, 15).

(A \ B )∪ C = A \ (B∩ C )for any three sets A,B and C Is the statement true or false? Give justification in support of your answer.

GreenRecycle.com is a recycle firm whose main business is recycling paper. There is a particular standard to grade the recycled papers in the firm, which grades the papers as P11, P12, P13, P21, P22, P31 and P32. a) Model the set of …

Minimize the following problems using the Karnaugh maps method. Z = f (A, B, C) = + B + AB + AC

Using the Karnaugh map method obtain the minimal sum of the products and product of sums expressions for the function F (A, B, C, D) = Σ (1, 5, 6, 7, 11, 12, 13, 15).

Let A = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10} and B = {1; 2; 3; 4}. Let R be the relation on P (A) de fined by: For any X,Y element in P(A), XRY if and only if X-B = Y-B. How many equivalence classes are there? Explain

(p->(q^r))^(not p->not q^not r))

Draw a Venn diagram of sets B,A and C where A . ⊆ ,B A ∩ C ≠ φ, B∩ C = φ What is the universal set you have chosen? Justify your choice of sets in the diagram.

Define (a) Graph; (b) Null graph; (c) Isolated vertex; (d) Pendant vertex; (e) Pseudo-graph; (f) Directed graph; (g) Adjacent nodes; (h) Incident edges;

Let A be an 8*8 boolean matrix (i.e.,every entry is 0 or 1).if the sum of the entries in A is 51, prove that there is a row i and a column j in A such that the entries in row i and in column j add up to more than 13. Further show that…

Which of the following statements are true and which are false? Give reasons for your answer. i) (∼ p∨q) and (p ← q) are logically equivalent. ii) (m+n)! > m!+n! for m, n any positive integers. iii) The number of distributions of m in…

a) Which of the following sentences are statements? Give reasons for your answer. i) Are the buses running today? ii) What a pleasant weather! iii) x2 +1 = 0 for a real value of x. iv) Every odd number is a prime. b) Let t(n) denote t…

a) Let Fn denote the nth Fibonacci number. Show by induction that every natural number is expressible as a sum Fn1 +Fn2 +···+Fnk where ni −ni+1 > 1 for each i ≥ 1. b) Find the number of ways of tying up 7 different books into 4 bundle…

a) Make a table of the values of the Boolean function f(x1,x2,x3) = x2 ⊕(x1 ∧x3) Write the function in DNF using the table. b) Find the general form of the solution to a linear homogeneous recurrence with constant coefficients for whi…

a) Let A be an 8×8 Boolean matrix (i.e. every entry is 0 or 1). If the sum of the entries in A is 51, prove that there is a row i and a column j in A such that the entries in row i and in column j add up to more than 13. Further, show…

A) Find the generating function of the recurrence an = 4an−1 −4an−2 +1 with initial conditions a0 = 1, a1 = 1. B) Express 3x4 +2x3 −2x2 +x in terms of [x]4, [x]3, [x]2 and [x]

Find the number of equivalence relations that can be defined on a set of 6 elements.

Explain 1. Game Tree 2. Decision Tree

Mathematical Induction. Suppose that you know that a cyclist rides the first kilometre in an infinitely long road, and that if this cyclist rides one kilometre, then she continues and rides the next kilometre. Prove that this cyclist …

Determine if the statements are valid arguments. I just got my ticket to the concert but if the number of tickets sold is more than 75% of its capacity, I will not want to go because it will be too crowded and I don’t like it. If my …

determine that wheter the functions from real numbers to real numbers are one to one f(n)=n^3 f(n)=n^2+1

In a lottery, players win a large prize when they pick four digits that match, in the correct order, four digits selected by a random mechanical process. A smaller prize is won if only three digits are matched. What is the probability…

If we reduce the number of elements by two, the number of permutations reduces thirty times. Find the number of elements.

How many 8-bits sequences that start with the same two bits or their fourth and fifth bits are equal or end with the same two bits are there?

Let D = {1, 2, 3}. The domain of the variables x and y will be D. Give an example of a predicate P(x, y) such that ∀x ∃y P(x, y) is true, but ∃y ∀x P(x, y) is false

Let D = {1, 2, 3}. The domain of the variables x and y will be D. Is it possible to find a predicate P(x, y) such that ∃y ∀x P(x, y) is true but ∀x ∃y P(x, y) is false? Explain

Determine if the following argument is valid using truth tables. p ∧ q → r __________ ∴ q → r

Determine if the following argument is valid using truth tables. p −→ r r _______ ∴ p

Determine if the following argument is valid using truth tables p ←→ q p∧ ∼ q _________ ∴ r

Determine if the following argument is valid using truth tables p → q p → r __________ ∴ p → (q ∧ r)

Determine if the following argument is valid using truth tables. p → (q ∨ r) ∼ q __________ ∴ p → r

Let D = {1, 2, 3}. The domain of the variables x and y will be D. Give an example of a predicate P(x, y) such that ∀x ∃y P(x, y) is true, but ∃y ∀x P(x, y) is false

Determine if the following argument is valid using truth tables. p ∧ q → r __________ ∴ q → r

Let D = {1, 2, 3}. The domain of the variables x and y will be D. Is it possible to find a predicate P(x, y) such that ∃y ∀x P(x, y) is true but ∀x ∃y P(x, y) is false? Explain

write the following boolean expressions in an equivalent sum of product canonical form in three variables x1, x2, and x3: 1. x1*x2 ? 3. (x1+X2)'*X3

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…

In how many ways can one select 7 member committee from 10 distinct persons if only three persons qualify to be chairperson?

Prove that the conditional proposition and its contrapositive are logically equivalent suing the truth table.

Find a counter-example to the following statement: For all real numbers x > 1, 1/x^2+1 ≤ 1/2^x+1

Prove that for all integers a, b, c if a|b then ac|bc

Prove that for all integers a, b, c such that c =/= 0, if ac|bc then a|b.

Prove that for all integers n, n(n + 2)(n + 4) is divisible by 3.

Draw the graph represented by the adjacency matrix . (0 0 1 0 1 0 1 0 0)

In how many ways can one select 7 member committee from 10 distinct persons if only three persons qualify to be chairperson?

Prove by mathematical induction the formula (1^3+2^3+3^3+4^3+....n^3)=(n^2(n+1)^2)/4

Prove that the conditional proposition p->q and its contrapositive ~q ->~p are logically equivalent using the truth table.

Given that p and q are propositions construct the truth table of p -> q p <-> q

Prove that the conditional proposition p⟶q and its contrapositive ~q⟶~p

Given that p and q are propositions construct the truth table of p⟶q p<->q

Recall that a real number x is rational if x = p/q for integers p, q with q = ̸= 0. Prove that if x is rational then 1/(2x+1) is rational. Then prove that if 1/(2x+1) is rational then x is rational.

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…

6.which of the following statement is true a.(p∨q)∧(p∨r)=p∨(q∧r) b.(p∧q)∧(p∨r)=p∨(q∧r) c.(p∧q)∨(p∨r)=p∨(q∧r) d.∼(p∨q)=∼(p∧∼q) 7. ____ reads â€œthe goods are standard if and only if the goods are expensiveâ€ a.p↔q b.∼p∧q c.∼∼q…

b) Given that p and q are propositions construct the truth table of; (4 Marks) i. p->q ii. p<->q

(p∧q)=(q∧p) and (p∨q)=(q∨p) implies an.............. Answers Options a. Associative laws b. Distributive Laws c. Commutative Laws d. Idempotent Laws

Draw the graph represented by the adjacency matrix . (001 010 100)

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…

A pair of dice is loaded. The probability that a 4 appears on the first die is 2/7, and the probability that a 3 appears on the second die is 2/7. Other outcomes for each die appear with probability 1/7. What is the probability of 7 appearing as the sum of the numbers when the two dice are rolled?

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Question ID: mtid-5-stid-8-sqid-106-qpid-35