Solved: 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 module today, then you will not play ML today. If you cannot play ML today, you can play ML tomorrow. Therefore, you read your module today, then you will play ML tomorrow. PROPOSITIONAL LOGIC: Let p, q, and r denote 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 an English sentence. (3 pts each) a. r ^ q → p b. p̅ → q c. q → p̅ v r 2. Write T if the above item is true and F if it is false. Show solution. (3 pts each) a. b. c.

Use algebra of sets to prove that, [(𝐵 − 𝐴)' ∩ 𝐴] − 𝐴' = 𝐴

(4). Construct a difference table to predict the next term of each sequence. (a) 9, 4, 3, 12, 37, 84, ... (b) 10, 10, 12, 16, 22, 30, ... (7). Use Polya’s four-step problem-solving strategy and the problemsolving procedures to …

.Find these values. a) Ã ¢ 1.1Ã ¢ b) Ã ¢ 1.1Ã ¢ c) Ã ¢ Ã ¢ 0.1Ã ¢ d) Ã ¢ Ã ¢ 0.1Ã ¢ e) Ã ¢ 2.99Ã ¢ f ) Ã ¢ Ã ¢ 2.99Ã…

Let X={1,2} and Y={a,b,c}. List the elements in each set.

. Translate in two ways each of these statements into logi- cal expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. …

Prove that for all integer n>=3, P(n+1,3) - P(n,3) = 3P(n,2)

If ad-bc=0 then: a/c = b/d y = a/c = b/d So, y is a constant and is not one-to-one. Finding the inverse function: Let y = f(x) = (ax+b) / (cx+d) So y (cx+d) = ax+b So cxy + dy = ax+b So cxy - ax = b - dy So x (cy - a) = b …

Find the domain and range of the following function given by 𝑓(𝑥)=√(3𝑥−5)(𝑥+4)𝑥3−16𝑥.

Show that if 𝑎𝑑−𝑏𝑐 ≠ 0, then the function 𝑓(𝑥)=𝑎𝑥+𝑏/𝑐𝑥+𝑑 is one-to-one and find its inverse.

Prove that ( ) ( ) 1, 1 , nPnnPnn  − − = .

Show that ( ) ( ) ( ) 1, , 1 , CnkCnkCnk + = − + .

(a)How many bit strings of length 8 contain at least 6 ones? (b)How many bit strings of length 8 contain at least 3 ones and 3 zeros?

If n fair six - sided dice are tossed and the numbers showing on top are recorded, how many (a) record sequences are possible? (b) sequence contain exactly one six? (c) sequences contain exactly four twos, assuming 4 n  …

Let A={n∈N:20≤n<50} and B={n∈N:10

Consider the setsAandB,whereA={3,|B|}andB={1,|A|,|B|}.Whatare the sets?

(4) Construct a difference table to predict the next term of each sequence. (a) 9, 4, 3, 12, 37, 84, ... (b) 10, 10, 12, 16, 22, 30, ..

In a group of computer engineering students, some like algebra and others do not like algrebra. There is no student in this group who is indifferent or has no opinion. Two students are randomly selected from this group. (a) How many o…

Question#1 Use algebra of sets to prove the following: i. (𝐵 − 𝐴) ∪ (𝐶 − 𝐴) = (𝐵 ∪ 𝐶) − 𝐴 ii. [(𝐵 − 𝐴) c∩ 𝐴] − 𝐴c= 𝐴 iii. (𝐴𝑐 ∪ 𝐵)c ∩ Ac= ∅ Question#2 Use Mathematical induction to prove the following generalization of one of…

Find out reflexive, symmetric and transitive closure of following relation R. R = {(1,2), (2,3), (3,3)}

1. Write the multi-sets of prime factors of given numbers. a. 150 b. 450 c. 1250 2. Find the cardinalities of each multiset in parts 2-1. 3. Present the application offset and multiset in software engineering? Give specific progr…

1. Find the theta notation of the following: a. 6n ³+ 12n ² + 1, for nÃ ¢ ¥1 b. 3n ²+ 2n log2 n, for nÃ ¢ ¥1 2. Find the theta notation for the number of times the statement x:=x + 1 is executed. a. for i:=1 to 2n do…

Q#1 For any sets 𝐴, 𝐵 and 𝐶, if 𝐴 ⊆ 𝐵, then 𝐴 ∩ 𝐶 ⊆ 𝐵 ∩ 𝐶. Q#2 For any sets 𝐴, 𝐵 and 𝐶, (𝐴 × 𝐶) ∩ (𝐵 × 𝐷) = (𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷). Q#3 Given sets 𝐴, 𝐵 and 𝐶, prove that 𝐴 × (𝐵 ∩ 𝐶) = (𝐴 × 𝐵) ∩ (𝐴 × 𝐶) Q#4 Prove that If 𝐴 and 𝐵 are se…

Refer to the relation R on the set {1,2,3,4,5) defined by the rule (x, y) = R if 3 divides x - y 1. List the elements of R 2. List the elements of R-1 3. Find the domain of R 4. Find the range of R 5. Find the domain of…

Find all the minimal and maximal elements of A= {2 ,3 ,4 ,6 ,8 ,24 ,48} with partial order of divisibility using Hasse diagram

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 …

Use deductive reasoning to show that the following procedure always produces a number that is equal to the original number. Procedure: Pick a number. Multiply the number by 6 and add 8. Divide the sum by 2, subtract twice the orig…

There are 4 adults and 6 children sitting around a round table. If there must be at least one child between any two adults, then how many ways are there for them to sit around the table? Rotations are considered the same, while re…

(a) How many bit strings of length 8 contain at least 6 ones? (b) How many bit strings of length 8 contain at least 3 ones and 3 zeros?

(a) Prove that for all integer n Ã ¢ ¥ 3, P (n +1,3) - P(n,3) = 3P(n,2) b)Prove that n.P(n-1,n-1) = P(n,n) c)show that C(n+1,k) = C (n,k-1) + C(n,k)

If n fairsix-sided dice are tossed and the numbers showing on top are recorded, how many (a) record sequences are possible? (b) sequence contain exactly one six? (c) sequences contain exactly four twos, assuming n ≥ 4?

Write an algorithm that outputs the smallest element in the sequence: SÃ ¢ , S2... Sn Use the procedure name find_small and output small. Make a recursive algorithm for finding the greatest common divisor of any non-negative i…

Let S be the following ordered sequence of elements S = {1, 2, 3, 4, 5, 6} and the universal set U be {1, 2, 3, 4, 5, 6}. Write down the characteristic vectors of • A = {1, 2, 4, 5}; • B = {3, 5}; • ∅; • A ∪ B; • A ∩ B; • A∪ ∼ …

Determine the number of subsets of size k of the set {1, 2, . . . , n} which do not contain consecutive integers. For instance, when n = 4 and k = 2, there are 3 such subsets, namely {1, 3}, {1, 4} and {2, 4}.

Determine the number of subsets of size k of the set {1, 2, . . . , n} which do not contain consecutive integers.

Determine the number of bijective functions f from {1, 2, . . . , 8} to itself such that f(i) 6= i for any even number i.

Find a system of distinct representatives for the following sets: {a, c, f}, {b, g}, {c, d, h}, {a, d, j}, {e, i, k}, {f, j, l}, {b, i, k}, {e, g}, {a, c, f, h, l}, {g, k}, {b, e, i}, {d, h, j}.

Find a system of distinct representatives for the following sets: {b,e,i,l}, {a,j,m}, {c,f,k}, {b,h,i,l}, {d,g,m}, {e,h,k,l}, {a,d,j}, {g,j,m}, {c,e,k}, {a,g,j}, {f,h,i}, {d,j,k,m}, {b,c,f}.

Fourteen chess players a, b, c, d, e, f, g, h, i, j, k, l, m, n are to be paired so they can play 7 games starting at the same time. The only pairs allowable are the following: {c,f}, {a,d}, {b,g}, {e,h}, {a,i}, {a,j}, {a,k}, {a,l}, {…

The rational numbers with 17 as a denominator are ... a) finite b) countably infinite c) uncountable d) non of these Choose 1 answer

Show that x2 is not O(x*log(x))

a particular algorithm increases in time as the number of operations n increases. Suppose the time complexity of this algorithm is given by: f(n)=4n2+5n2*log(n) Show that f(n) is O(g(n)) for g(n) = n3

A particular algorithm increases in time as the number of operations, n, increases. Suppose the time complexity of this algorithm is given by: f(n) =4n3+5n2 * log(n). Show that f(n) is O(g(n)) for g(n) = n3.

Solve the congruence 5x≡1(mod12) Hint: 0≤x≤11

f(n)=4n 2 +5n 2 ∗log(n) , g(n) = n^3g(n)=n 3 Now to show f(n)=O(g(n))f(n)=O(g(n)) when number of operation increases i.e., n\rightarrow \inftyn→∞ then \frac{f(n)}{g(n)}=\frac{4n^3+5n^2.log\ n}{n^3}=4+…

a particular algorithm increases in time as the number of operations n increases. Suppose the time complexity of this algorithm is given by: f(n)=4n2+5n2*log(n) Show that f(n) is O(g(n)) for g(n) = n3

A particular algorithm increases in time as the number of operations, n, increases. Suppose the time complexity of this algorithm is given by: f(n) =4n3+5n2 * log(n). Show that f(n) is O(g(n)) for g(n) = n3.

Solve the congruence 5x≡1(mod12) Hint: 0≤x≤11

Suppose an = 2n - 2*an-1 and a0=0 What is a2n?

Let R5 = {(1,2), (1,4), (1,6), (2,2), (2,4), (2, 6), (3,2), (3,4), (3,6), (4,2), (4,4), (4,6), (5,2), (5,4), (5,6)} and R6 = {(1,2), (1,4), (1,6), (2,4), (2,6), (3,6)} be relations. How many ordered pairs are there in R5 U R6?

Draw a truth table for the logic statement (R ∧ ¬T) → (S ∨ T), where R, S, T are Boolean sub-statements.

𝑈 = {𝑥|𝑥 ∈ 𝑅 ∧ 𝑥 > 14} 𝐴 = [−2, 25), 𝐵 = [15, 30]. Find: a) 𝐴̅̅̅∪̅̅̅𝐵̅ =?

A = {0, 2, 4, 6, 8, 10} B = {0, 1, 2, 3, 4, 5, 6} C = {4, 5, 6, 7, 8, 9, 10} a. A\cap B\cap CA∩B∩C, requires us to determine the elements common to set A, set B and set C. Therefore, A\cap B\cap C=\{4,6…

Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find a) A ∩ B ∩ C. b) A ∪ B ∪ C. c) (A ∪ B) ∩ C. d) (A ∩ B) ∪ C.

Let A, B, and C be sets. Show that a) (A ∪ B) ⊆ (A ∪ B ∪ C). b) (A ∩ B ∩ C) ⊆ (A ∩ B). c) (B − A) ∪ (C − A) = (B ∪ C) − A.

Find the truth set of each of these predicates where the domain is the set of integers. a) P(x): 𝑥 3 ≥ 1 b) Q(x): 𝑥 3 = 8 c) R(x): x < 𝑥 2 d) F(x): |x +5| < 52

What is the cardinality of each of these sets? a) {a, 0, {a, 0}} b) {{a}} c) {∅, a, {a}} d) {0, 1, a, {a}, {a, {a}}}

Use set builder notation to give a description of each of these sets. a) {0, 3, 6, 9, 12} b) {−3,−2,−1, 0, 1, 2, 3} c) {m, n, o, p}

Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of these sets are subsets of which other of these sets.

Determine whether these statements are true or false. a) ∅ ∈ {∅} b) ∅ ∈ {∅, {∅}} c) {∅} ∈ {∅} d) {∅} ∈ {{∅}} e) {∅} ⊂ {∅, {∅}} f ) {{∅}} ⊂ {∅, {∅}} g) {{∅}} ⊂ {{∅}, {∅}}

Let A = {m, n, o}, B = {a, b}, and C = {7, 2}. Find: a) A x B x C. b) C x B x A. c) C x A x B. d) B x B x B.Let A = {m, n, o}, B = {a, b}, and C = {7, 2}. Find: a) A x B x C. b) C x B x A. c) C x A x B. d) B x B x B.

Draw De Morgan’s Law, Absorption and Complementation Laws

If n fair six-sided dice are tossed and the numbers showing on top are recorded, how many (a) record sequences are possible? (b) sequence contain exactly one six? (c) sequences contain exactly four twos, assuming n  4 ?

We are given two functions f : A → B and g : B → C. Prove that if f and g are onto, then g ◦ f is onto.

For each of the following relations, determine whether they are reflexive, symmetric, anti- symmetric, and/or transitive, and give a brief justification for each property. a) R ⊆ Z × Z where xRy iff x = y 2 b) The empty relation: R ⊆ …

Let A={1,2,3,4,5} and R be the relation on A such that R={(1,1),(1,4),(2,2),(3,4),(3,5),(4,1),(5,2),(5,5)}. Find the transitive closure of R by Warshall's Algorithm

Let A={1, 2, 3, 4} and R be the relation on A such that R = { (1, 2), (1, 4), (2, 1), (2, 3), (3, 1) }. Find the transitive closure of R by warshall's algorithm.

If n fair six-sided dice are tossed and the numbers showing on top are recorded, how many (a) record sequences are possible? (b) sequence contain exactly one six? (c) sequences contain exactly four twos, assuming n >= 4

Prove that n*P( n -1,n - 1) = P (n, n)

Show that C (n+1, k) = C (n, k -1) + C (n, k)

Prove that using proof by contradiction. √2 + √6 < √15

Use a truth table to verify this De Morgan’s law: ¬(p ∧ q) ≡ ¬p ∨ ¬q

Determine whether each set below 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 oneto-one (Injective), onto (Surjective), or both…

Determine whether this function f(n) = n2 − 1 is one-to-one, onto, or both. Explain your answers. The domain of each function is the set of all integers. The codomain of each function is also the set of all integers

Let f be the function from x ={0, 1, 2, 3, 4, 5} to X defined by f(x) = 4x mod 6. Write f as a set of ordered pairs and draw the arrow diagram of f . Is f one-to-one? Is f onto?

a) Simplify the following Boolean expression using three variable maps F(x, y, z) = x y z + x’y’ z + x y’ z’ b)Simplify the following Boolean expression using four variable maps: F(w,x, y, z) = Σ(0,1,4,5,6,7,8,9)

Q no.1) Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Draw the Venn diagrams for each of these combinations of the sets A, B, and C. a) A ∩ (B ∪ C) b) A ∩ B ∩ C

Exercise # 3 a. Ten persons have first names Alice, Billy, and Charlie and last names Lee, Mendoza and Navarro. Show that at least two persons have the same first and last names. b. In any group of 367 people, at least two people must…

Exercise # 2 A committee composing of six members naming Ana, Ben, Cora, Dina, Elphie and Francis is to select a chairperson, secretary and a treasurer. d. In how many ways can they select the chairperson, secretary and treasurer? e. …

1. Find the sets A and B if A − B = {1, 5, 7, 8}, B − A = {2, 10}, and A ∩ B = {3, 6, 9}. 2. If B and C are two sets, prove or disprove the identity, B×C=C×B. 3. Prove that if A and B are sets, then A ∩ (A ∪ B) = A. 4. Suppose that A …

Determine whether each of these functions is a bijection from R to R. a)f (x) = x^3

Using mathematical induction, prove that for every positive integer n, 1 · 2 + 2 · 3 + · · · + n(n + 1) = (n(n + 1)(n + 2)) / 3

Let fn be the nth Fibonacci number. Prove that, for n > 0 [Hint: use strong induction]: fn = 1/√5 [((1+√5)/2)n - ((1-√5)/2)n ]

There is a long line of eager children outside of your house for trick-or-treating, and with good reason! Word has gotten around that you will give out 3k pieces of candy to the kth trick-or-treater to arrive. Children love you, denti…

Let the sequence Tn be defined by T1 = T2 = T3 = 1 and Tn = Tn-1 + Tn-2 + Tn-3 for n ≥ 4. Use induction to prove that Tn < 2n for n ≥ 4

For the following recursive functions, Prove that am,n = m · n. Here am,n is defined recursively for (m, n) ∈ N × N. am.n = { 0 if m=0 n+am-1,n if m!=0

Exercise #1: Restaurant Menu Appetizers Nachos …………………… 120.00 Salad ……………………… 125.00 Main Courses Hamburger ………………. 125.00 Cheeseburger …………… 130.00 Fish Filet …………………. 155.00 Beverages Tea ……………………….. 120.00 Milk ………………………. .120.00 …

N=2^{12}=4096N=2 12 =4096 ways How to review the solution in polya strategy

Show, by giving a proof by contradiction, that if 100 balls are placed in nine boxes, some box contains 12 or more balls.

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Question ID: mtid-5-stid-8-sqid-3109-qpid-1808