Solved: Use Polya’s four-step problem-solving strategy and the problem- solving procedures to solve each of the following exercises. (a) An android phone and a Bluetooth speaker together cost Php6500.00 . The phone costs Php2000.00 more than the speaker. What is the cost of the phone? (b) In how many ways can you answer a 12-question true-false test if you answer each question with either a “true” or a “false”? (c) A total of 364 first-year students of CMU are currently enrolled in the subject GEC14. If there are 26 more girls than boys, how many girls are there?

Determine the truth value of each of these statements if the domain consists of all integers. NOTE: Explain how did you get the truth value on each statements in your own words. a) ᴲxP(2n=3n) b) ⱯxP(3n=<4n)

Here are some scenarios: According to market research, a business has a 75% chance of making money in the first 3 years According to lab testing, of a certain kind of experimental light bulb will work after 3 years. According to expe…

PREDICATE LOGIC: Write the following predicates symbolically and determine its truth value. Note: Use at least three (3) values for the variables. 1. for every real number x, if x>1 then x – 1 > 1 2. for some real number x, x2 ≤ 0…

Provide the answers to the following items. (5 items x 3 points) 1. Given A = (2, 4, 6, 8} and B = {3, 4, 5, 6}, determine: a. A ∪ B b. A ∩ B 2. Given A = {3, 5, 7, 9} and B = {4, 5, 6, 7}, determine: a. A - B b. B - A c. …

1. Given A = (2, 4, 6, 8} and B = {3, 4, 5, 6}, determine: a. A ∪ B b. A ∩ B 2. Given A = {3, 5, 7, 9} and B = {4, 5, 6, 7}, determine: a. A - B b. B - A c. A ∩ B

Let p denote He is rich and let q denote He is happy. Write each statement in symbolic form using p and q. Note that He is poor and He is unhappy are equivalent to ¬p and ¬q, respectively. (a) If he is rich, then he is unhappy. (c) …

Let S = ℤ+, a ~ b if a − b is divisible by 2. Is S an equivalence relation?

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 get high scores. The…

Determine if the following argument is valid. Explain by using rule of inference. (5 pts each) 1. If you perform every programming problem in the module, then you will learn programming. You learned programming. Therefore, you per…

What is the symbolic form of the statement, “No one in this class is wearing pants and a guitarist” if the domain of x is persons in this class, A(x): x is wearing pants and B(x): x is a guitarist?

PROPOSITIONAL LOGIC: A. Let p, q and r denotes the following statements: p: A square has four equal side q: Rectangle has 2 parallel sides r: A square is a rectangle. 1. Express each of the following into English sentence. a. r …

PREDICATE LOGIC.(25 pts) A. Let P(x) be the statement x 2 > x4. If the domain consists of the integers, what are the truth values? 1. P(0) 2. P(-1) 3. P(1) 4. P(2) 5. ∃xP(x) 6. ∀xP(x)

PREDICATE LOGIC: A.Write the following predicates symbolically and determine their true value. Note: Use at least three (3) values for the variables. 1. for every real number x, if x>1 then x – 1 > 1 2. for some real number x, x2 …

A. Write each statement into its symbolic form x: PJ is a mathematician y: MJ is a programmer a. PJ is not a mathematician. b. PJ is a mathematician while MJ is a programmer. c. If PJ is a mathematician then MJ is not a programme…

A. Show whether or not p → q ≡ (p ^ q) v (𝒑̅ ^ 𝒒̅) B.Let P(x) denote the statement 1 ------ x2+1>1. If its domain are all real numbers, what is the truth value of the following quantified statement? (5 pts each) 1. ∃xP(x) 2. ∀x…

Consider the following statements. Write each statement into its symbolic form.(2 pts each) Let: p: Ann solved the problem in discrete math. q: Bryan solved the problem in discrete math. r: Cris solved the problem in discrete math…

A. SET. Let A, B and C are sets and U be universal set. U = {-1, 0, 1, 2, 3, 4, 5, 6, a, b, c, d, e} A = {-1, 1, 2, 4} B = {0, 2, 4, 6} C = {b, c, d} Find for the following. Show complete solutions. (3 pts each) 1. 𝐵 ∪ 𝐶 2. 𝐴 …

Rewrite each sentence symbolically and determine the truth values. Write T if it is true and F if it is false. Show complete solution. (5 pts each) 1. For some integer x, 𝑥 = 𝑥2 − 2 2. For every real number x, 𝑖𝑓 𝑥2 − 1 > 𝑥 𝑡ℎ𝑒𝑛 𝑥…

Consider the following statements. Write each statement into its symbolic form.(2 pts each) Let: p: Ann solved the problem in discrete math. q: Bryan solved the problem in discrete math. r: Cris solved the problem in discrete math…

A. SET. Let A, B and C are sets and U be universal set. U = {-1, 0, 1, 2, 3, 4, 5, 6, a, b, c, d, e} A = {-1, 1, 2, 4} B = {0, 2, 4, 6} C = {b, c, d} Find for the following. Show complete solutions. 1. 𝐵 ∪ 𝐶 2. 𝐴 − 𝐵 𝑥 𝐶 3. 𝑃𝑜…

dx/dt=x-y dy/dt=x +3y

Apply your controller established in Step 1 to the original nonlinear model (1), and validate if the vehicle can still achieve any desired speed by simulations. Discuss the similarities and differences of the responses between t…

Consider the following statements. Write each statement into its symbolic form. Let: p: Ann solved the problem in discrete math. q: Bryan solved the problem in discrete math. r: Cris solved the problem in discrete math. s: Derynn…

PROBLEM SOLVING. A. SET. Let A, B and C are sets and U be universal set. U = {-1, 0, 1, 2, 3, 4, 5, 6, a, b, c, d, e} A = {-1, 1, 2, 4} B = {0, 2, 4, 6} C = {b, c, d} Find for the following. Show complete solutions. 1. 𝐵 ∪ 𝐶 2…

PREDICATE LOGIC. Rewrite each sentence symbolically and determine the truth values. Write T if it is true and F if it is false. Show complete solution. (5 pts each) 1. For some integer x, 𝑥 = 𝑥2 − 2 2. For every real number x, 𝑖𝑓 𝑥…

PROBLEM SOLVING. A. SET. Let A, B and C are sets and U be universal set. U = {-1, 0, 1, 2, 3, 4, 5, 6, a, b, c, d, e} A = {-1, 1, 2, 4} B = {0, 2, 4, 6} C = {b, c, d} Find for the following. Show complete solutions. (3 pts each)…

C. RELATION. Consider X = {-3, -2, -1, 0, 1} defined by (x,y) ∈ R if x≥ y. Find for: 1. Elements of R (3 pts) 2. Domain and Range of R (2 pts) 3. Draw the digraph (3 pts) 4. Identify the properties of R (2pts)

RULE OF INFERENCE. Determine if the following argument is valid. If it is valid, what rule of inference is used in each of the following arguments? Show solution. (4 pts each) 1. Joy wrote a C++ source code, or Jen wrote a Java sour…

MATHEMATICAL INDUCTION AND RECURRENCE Solve the following. (10 pts each) 1. Prove P(n) = n2 (n + 1) 2. Recurrence relation an = 2n with the initial term a1 = 2.

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…

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