Solved: Let p, q and r be the propositions p : Grizzly bears have been seen in the area. q : Hiking is safe on the trail. r : Berries are ripe along the trail. Write these propositions using p, q, and r and logical connectives (including negations). (a) Berries are ripe along the trail, but grizzly bears have not been seen in the area. (b) Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail. (c) If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area. (d) It is not safe to hike on the trail, but grizzly bears have not been seen in the area and the berries along the trail are ripe. (e) For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area. (f) Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries are ripe along the trail.

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 v…

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…

p → ~q

Suppose you have a graph with v vertices and e edges that satisfies v=e+1. Must the graph be a tree? Prove your answer.

Proof by contradiction that if n is a positive integer, then n is odd if and only if 5n + 6 is odd

Show that if x is an integer then x2+x-41= 0 produce prime numbers.

As you now that 03 0 13 1 23 8 33 27 43 64 53 125 so, we have a proof that every positive integer is the sum of the cubes of eight non negative integers. For example 7= 13 + 13 + 13 +13+13+13+ 13 +03 12= 23+ 13 + 13 + 13 +13+…

Show that if x is an integer then x2+x-41= 0 produce prime numbers

Using propositional logic, verify each of the following equivalences using basic equivalences 1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S 2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

Use the truth table to transform each of the following wffs into the full conjunctive normal form (P→Q)→P. P→(Q→P) (P∨Q)∧R. P→Q∧R. Q∧¬P→P

Represent the following relation to a directed graph. relation R = {(1, 1), (1,2), (3, 2)} on set S = {1,2,3},}

RULE OF INFERENCE A. What rule of inference is used in each of the following arguments? Show solution. (5 pts each) 1. If I will read my modules, then I can answer all the activities. If I can answer all the activities, then I will…

What rule of inference is used in each of the following arguments? Show solution. 1. If it will rain today, then the classes are suspended. The classes are not suspended today. Therefore, it did not rain today. 2. If you read your m…

Draw the Hasse diagram for divisibility on the set, {1, 2, 3, 4, 5, 6, 7, 8}?

using truth tables, verify each of the following equivalences using basic equivalences 1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S 2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

Give a contrapositive proof of the theorem; "If n is an interfer and 3n + 2 is even, then n is even."?

Prove by induction that P(n): 2+3+3...+n=n(n+1)/2 Æn ≥ 1

Generate up to the seventh and nineth rows of the Pascal triangle.

verify each of the following equivalences using basic equivalences 1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S 2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

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

Let 𝑆 = {𝑎, 𝑏, 𝑐} and 𝑅 = {(𝑎, 𝑎), (𝑏, 𝑏), (𝑐, 𝑐), (𝑏, 𝑐), (𝑐, 𝑏)}, find [𝑎], [𝑏] and [𝑐] (that is the equivalent class of a, b, and c). Hence or otherwise find the set of equivalent class of 𝑎, 𝑏 and 𝑐?

Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where: 𝑅 = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has:…

Give an indirect proof of the theorem; “If 𝑛 is an integer and n3 +13is odd, then n is even.”?

For the following Boolean functions, give a tabular representation of their Boolean expressions: (i) F x y z( , , ) =[x y z  ] (ii) F x y z( , , ) =[x z y z  ] [ ] What can you say about (i) and …

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.

In the following argument, determine the validity or otherwise of the Statement: b) “If you play football during a thunderstorm, you’ll get hit by lightning. You didn’t get hit by lightning. Therefore, you didn’t pla…

If 𝐴 and B are finite sets which are subsets of 𝑈. Establish a formula for 𝑛(𝐴 ∪ 𝐵) in terms of 𝑛(𝐴), 𝑛(𝐵) and 𝑛(𝐴 ∩ 𝐵). Hence or otherwise deduce a formula for a particular case where A and B are disjoint?

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 …

Draw the Hasse Diagram representing the partial ordering {(a, b) | a divides b} on {1, 2, 3, 4, 5, 6, 10, 15, 20, 30, 60}?

Find the negation of each: 1. ∃ x ( (x+3=2) ^ (x - 2 =1))

TRUTH TABLES: MAKE A TRUTH TABLE FOR EACH OF THE FOLLOWING STATEMENT 1. p ∧ q 3 points 2. p ∨ q 3 points 3. p∨~q 4 points

In how many ways the word scooter be arranged when repetition is not allowed.

Consider set s={1,2,3,9,6,18,} and the relation <= is x divides y,then draw the hasse diagram for Posey(s,<=)?

determine whether {2} and 2 is an element of {{2}, {2,{2}}

Inclusion-Exclusion Principle A group of 191 students, which are taking French, business and music; 36 are taking French and business; 20 are taking French and music; 18 are taking business and music; 65 are taking French, 76 are t…

Draw the Hasse Diagram representing the partial ordering {(a, b) | a divides b} on {1, 2, 3, 4, 5, 6, 10, 15, 20, 30, 60}?

For the Boolean functions F1 and F2 below, write down the corresponding (i) Boolean expression in its disjunctive normal form; (ii)Using the Karnaugh map method, write down the simple Boolean expression? …

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…

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Question ID: mtid-5-stid-8-sqid-2617-qpid-1087