Solved: 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 ^ q → p b. p̅ → q c. q → p̅ v r 2. Write T if the above item is true and F if it false. Show solution. (3 pts each) a. b. c B. Show whether or not p ↔ q ≡ (p → q) ^ (q → p) C. Find the converse, inverse and contrapositive of the implication: “If today is Monday then, I have an exam today.” (3 pts each) 1. Inverse: 2. Converse: 3. Contrapositive:

(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.

Show by giving a proof by contrapositive, that if 3n+2 is odd, then n is odd

a n+1 −a n =3n characteristic equation: 1/x-1/x^2=01/x−1/x 2 =0 x-1=0x−1=0 x=1x=1 homogeneous solution: a_h=cx^n=ca h =cx n =c particular solution: a_t=An^2+Bn+Ca t =An 2 +Bn+C A(n+1)^2+B(n+1)+C-A…

Write these propositions using p and q and logical connectives (including negations). a) It is below freezing and snowing. b) It is below freezing but not snowing. c) It is not below freezing and it is not snowing. d) It is either sno…

Find ⋃ 𝑨𝒊 ∞ 𝒊=𝟏 and ⋂ 𝑨𝒊 ∞ 𝒊=𝟏 where: 𝑨𝒊 = {𝒊,𝒊 + 𝟏,𝒊 + 𝟐, … } for every positive integer 𝒊.

Given that n is a positive integer and in the expansion of (1+ax)n, the coefficient of x is 200. Find the coefficient of x in the expansion of (1+ax)2n.

Q1 Let R = {(1,4),(2,1),(2,5),(2,4),(4,3),(5,3),(3,2)}. Use warshall’s algorithm to find the matrix of transitive closure where A = {1, 2, 3, 4, 5}

A class has 175 students. The following data shows the number of students taking one or more subjects. Mathematics 100, Physics 70, Chemistry 40; Mathematics and Physics 30, Mathematics and Chemistry 28, Physics and Chemistry 23; Math…

Let 𝐴={1,2,3,4}. Determine the truth value of each statement: i. ∀𝑥 ∈𝐴,𝑥+3<6

i. false: 4+3>64+3>6 ii. true: 1+3<61+3<6 iii. false: 2\cdot1^2+1=32⋅1 2 +1=3 2\cdot2^2+2=102⋅2 2 +2=10 2\cdot3^2+3=212⋅3 2 +3=21 2\cdot4^2+4=362⋅4 2 +4=36

Let 𝐴={1,2,3,4}. Determine the truth value of each statement: i. ∀𝑥 ∈𝐴,𝑥+3<6 ii. ∃𝑥,𝑥+3<6 iii. ∃𝑥,2𝑥2 +𝑥 =15

Let Q(x)Q(x) be the statement “x + 1 > 2xx+1>2x ”. If the domain consists of all integers, let us find the following truth values. a) Since 1>0,1>0, we get that 𝑄(0)Q(0) is true. b) Taking into account that it is not true that 0…

Let Q(x) be the statement “x + 1 > 2x.” If the domain consists of all integers, what are these truth values? (7 pts.) a) 𝑄(0) b) 𝑄(−1) c) 𝑄(1) d) ∃𝑥𝑄(𝑥) e) ∀𝑥𝑄(𝑥) f) ∃𝑥¬𝑄(𝑥) g) ∀𝑥¬𝑄(𝑥)

20. Determine the truth value of each statement if the domain consists of all real numbers. (4 pts.) a) ∃𝑥(𝑥3 = −1) b) ∃𝑥(𝑥4 < 𝑥2) c) ∀𝑥((−𝑥)2 = 𝑥2) d) ∀𝑥(2𝑥 > 𝑥)

prove that n.P(n-1,n-1)=p(n,n)

Let R = {(1,4), (2,1), (2,5),(2,4),(4,3),(5,3),(3,2)} on the set A = {1, 2, 3, 4, 5}. Use Warshall’s algorithm to find transitive closure of R.

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

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?

Which of the following functions are injective? Which are surjective? a) f: Z → Z given by f(x) = x2 + 1. b) g: N → N given by g(x) = 2x. c) h: R → R given by h(x) = 5x - 1.

If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area.

Create a schematic diagram of all odd numbers from 20 to 45

(a) If x is nonnegative, then x is positive or x is 0.

4. v defined by vn = n! + 2, n ≥ 1. (a) Find v3 (b) Find Σ 4 on top , i=1 at the bottom and Vi on the right hand side of Sigma (c) Is v increasing, decreasing, non-increasing or non-decreasing?

1.b is defined by bn = n(−1)n, n ≥ 1 (a) Find Σ when 4 is on top, i=1 at the bottom and bi at the right hand side (b) Is b increasing, decreasing, non-increasing or non-decreasing?

Given ϕ2 = 5. Is ϕ increasing, decreasing, non-increasing and/or non-decreasing?

Find all substrings of the string babc.

1.Draw the digraph of the following relations. (a) R = {(1, 2), (2, 3), (3, 4), (4, 1)} on {1, 2, 3, 4} (b) R={(1, 2), (2, 1), (3, 3), (1, 1), (2, 2)} on X = {1, 2, 3}

For each relation below, determine if they are reflexive, symmetric, anti-symmetric, and transitive. (a) X= { 1, 2, 3, 4} R1={(1, 2),(2, 3),(3, 4)} (b) X = {a, b, c, d, e} R1 = { (a, a), (a, b), (a, e), (b, b), (b, e), (c, c),…

A discrete mathematics class contains I mathematics major who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors who are sophomores. 2 mathematics majors who are juniors. 2 computer science majors …

Use quantifiers and predicates with more than one vari- able to express these statements. a) There is a student in this class who can speak Hindi. b) Every student in this class plays some sport. c) Some student in this class has …

Create a schematic diagram of all odd numbers from 20 to 45

State whether the following graph g1 and g2 are isotropic or not

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Question ID: mtid-5-stid-8-sqid-3143-qpid-1842