Solved: 1. Expand the following using the Binomial Theorem a. (x-3y)^3 b. (2a+b^-3)^4 c. (a+b+c)² 2. Solve the recurrence relation Pn = Pn-1 + n with the initial condition p_1 = 2 using iteration.

Show that the relation R=∅ on the empty set S=∅ is reflexive, symmetric, and transitive.

(a) Find the number of relations on the set S={a, b, c, d, e}? (b) How many relations are there on the set S={a, b, c, d, e} that contain (a, a) and (b, b)?

Let S be a set with 6 elements and let a and b be distinct elements of S. How many relations R are there on S such that... (a) (a, b) ϵ R? b) (a; b) ∉ S? (c) no ordered pair in R has a as its first element? (d) at least one ordered pa…

Let R be a reflexive relation on a finite set A, and let MR be the bit matrix representing R. Specify the value of the entries on the main diagonal.

Let R be a symmetric relation on a finite set A, and let MR be the bit matrix representing R. Is MR necessarily a symmetric matrix? Why or why not?

Use Mathematical Induction to show that if MR is the bit matrix representing the relation R, then M^[n]R is the matrix representing R^n. (This was how the question was stated. If you're confused about the terms M^[n]R and R^n, they ar…

Let S be the set of bit strings of length no larger than 6, and define an equivalence relation R on S as follows: (x, y) ϵ R if and only if x and y are of the same length. Specify the partition P of S that arises from R.

Let S ={a, b, c, d, e}, and P={{a, b},{c, d},{e}}. (a) Verify that P really is a partiton of S. (b) Find the equivalence relation R on S induced by P.

Suppose S is a set, and P={S} is a partition of S. Find the equivalence relation R corresponding to P.

Let p and q be distinct positive primes. Define the partial ordering ⪯ of S={1, p, p^2, p^3, p^4, q, q^2, q^3, q^4} as follows: a⪯b if and only if a|b. Draw the Hasse diagram for this partial ordering of S.

Let S={1,2,3,4}, and define a partial ordering of P(S) (the power set of S), by: A⪯B if and only if A⊆B. Is this partial ordering in fact a total ordering (chain)? Why or why not?

Suppose S is a set containing 5 elements, and that ⪯ is a total ordering of S. Draw the Hasse diagram for ⪯ (no need to label the vertices in your diagram).

How many distinct permutations are there of the letters of the word "MARRIAGE"? a.30240 b.40320 c.20160 d.10080

How many 4-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5 if the first digit must not be 0 and repetition of digits is not allowed? a.300 b.320 c.360 d.280

There are two webcams, X and Y that can automatically switch On or Off at any given time to record a live video streaming session. To ensure the smooth recording session, the live feed input system is programmed with certain criteria.…

Find the Transitive closure by using Warshall’s Algorithm where A= {1, 2, 3, 4, 5, 6} and R= {(x,y)| (x-y)=2}

From a set of numbers 1-15, how many numbers should be picked to form pair of numbers that adds up to 16? Illustrate the solution.

Suppose S is a set containing 5 elements, and that ⪯ is a total ordering of S. Draw the Hasse diagram for ⪯ (no need to label the vertices in your diagram).

Model two contextualized problems using binary trees both quantitatively and qualitatively.

Show graphically that which of the following is one to one function (1)f(x)=ln(x) (2)g(x)=e^x (3) h(x)=x³

(i) Give a formal definition of a graph G and its complement G. (ii) Draw the graph with degree sequence 4, 4, 4, 4, 4, 4 and show that it is unique. (iii) Suppose a graph G has (0, 1) adjacency matrix A(G) = A. Write down the adjacen…

There are two webcams, X and Y that can automatically switch On or Off at any given time to record a live video streaming session. To ensure the smooth recording session, the live feed input system is programmed with certain criteria.…

Prove that log2 3 is irrational.

How many numbers are divisible by 2, 5, 9, and 13 between 100 and 100,000? How to implement this question in discrete math ?

a) Let p, q, and r be the propositions p : You study hard. q : You are on merit r : You do not get degree from Iqra University. Express each of these propositions as an English sentence. i) ¬q ↔ r ii) q →¬ r iii) p∨q ∨¬r iv) (p →¬r)∨(…

b) What is the value of x after each of these statements is encountered in a computer program, if x =3 before the statement is reached? i) if x +2=5 then x := 3*x +5 ii) if (x +1=4) OR (2x +2=3) then x := x +1 iii) if (2x +3=5) AND (3…

Suppose that the domain of the propositional function P(x) consists of the integers 1, 2, 3, 4, and 5. Express these statements without using quantiﬁers, instead using only negations, disjunctions, and conjunctions. a) ∃xP(x) b) ∀xP(x…

a) We have the following set of information about a computer program, find the mistake in the program using Rules of Inferences. i. Either a variable is not declared or there is a syntax error in the fifth line. ii. If there is a synt…

1.Determine whether ¬(p∨(¬p∧q)) and ¬p∧¬q are equivalent without using truth table. 2.Determine whether the compound proposition ~(p∨q)∨(~p∧q)∨p to tautology. 3.Determine whether (p → q)∧(p → r)≡p → (q ∧r) using a truth table

Solve the following recurrence relations i) Fn= Fn-1 +Fn-2 where a1=a2=1 ii) an=2an-1 - an-2 +2 where a1 = 1 and a2 = 5

Use mathimatical Induction to prove that 1²+2²+3²+.......n²= n(n+1)(2n+1) divided by 6

We have the following set of information about a computer program, find the mistake in the program using Rules of Inferences. i. Either a variable is not declared or there is a syntax error in the fifth line. ii. If there is a syntax …

Definition of Function, Domain, Codomain and Range, Well defined function, Types of functions(Injective ,Surjective, Bijective, Composite)

Suppose S is a set containing 5 elements, and that ⪯ is a total ordering of S. Draw the Hasse diagram for ⪯ (no need to label the vertices in your diagram).

Transform by making change of variable j = i-4.

Consider the scenario of Knights and Knaves. You visited that island and met five people there. Suppose those were A, B, C, D, E [10]  A: B is a knight.  B: E is a knight.  C: A is a knave.  D: There are at most two knights a…

Determine the truth value of each of the quantified statements below if the domain consists of R. (a)∃x(x^5 = -1) (b)∃x(x^6< x^4) (c)∀x((-x)4=x^4) (d)∀x(2x > x)

(a) Let S be a set, and define the set W as follows: Basis: ∅ ϵ W.Recursive Definition: If x ϵ S and A ϵ W, then {x} U A ϵ W. Provide an explicit description of W, and justify your answer.

Define r(n, m) : N x Z+ -> N be the remainder obtained when dividing m into n. Define a function fm: N x N -> N as follows: f(n, k) = k if r(n, m) = 0, and f(n, k) = f(n-m, k+ 1) otherwise. Describe in terms of a single well-known ari…

For all parts of this problem, let S be a finite set, with |S|=n. (a) If f:S -> S is onto, does it necessarily follow that f is invertible? Why or why not? (b) Compute the number of distinct functions g:S -> S that are onto. (c) Find …

(a) Define r(n, m) : N x Z+ ->N be the remainder obtained when dividing m into n. Define the function g: Z+ x Z+ ->Z+ as follows: g(a, b) = b if r(a, b) = 0, and g(a, b) = g(b, r(a,b)) otherwise. Describe what g is calculating, and ju…

(b) Find a concise, explicit description for the set A, which is defined by 1 ∈ A, and if a and B are bit strings in A, then aB ∈ A, 0aB ∈ A, a0B ∈ A, and aB0 ∈ A.

(a) We say that two positive integers a, b are relatively prime if the greatest common divisor of a and b is 1. Let p, q be distinct positive prime numbers such that n=pq. Calculate the number of positive integers not exceeding n that…

b) Read Section 2.3.6 (on page 161) of the 8'th edition of Rosen's book on partial functions. Let A and B be finite sets, with |A| = m and |B| = n. Calculate the number of partial functions f: A -> B.

How many ternary strings (i.e., the only allowable characters are 0, 1, and 2) of length 15 are there containing exactly four 0s, five 1s, and six 2s?

How many solutions are there to the inequality x1+x2+x3+x4≤15, where x1,x2,x3, and x4 are nonnegative integers? Hint: Introduce an extra variable x5 and consider x1+x2+x3+x4+x5 = 15.

Suppose a friend asks you to list out all relations on the set S = {n|n ∈ Z+ and n≤1000}. Find the number of number of relations on S. Is your friend making a reasonable request?

Let R be a relation from A to B. Both sets are finite, with |A|=n and |B|=m. Define the complementary relation "R bar" as follows: R bar={(a, b)|(a,b)∈R} Calculate |R bar|.

(a) If A={1,2,3,4}, calculate the number of reflexive relations on A. (b) If B={1,2,3,4,5}, calculate the number of symmetric relations on B.

Let S be a finite set, with |S|=n. If the bit matrix MR representing R contains exactly r entries that are 0, how many entries of MRbar are 0? Note: "MRbar" is supposed to be just like MR, which you should know has R as a s…

Let S be a finite non-empty set. How many relations on S are simultaneously an equivalence relation and a partial order? Justify your answer.

Define a function A: N x N -> N as follows: A(m, n) ={2n, if m= 0; 0, if m≥1 and n= 0; 2, if m≥1 and n= 1; A(m-1, A(m, n-1)), if m≥1 and n≥2 (a) Calculate the following: (i)A(1,0) (ii)A(0,1) (iii)A(1,1) (iv)A(2,2).

Define a function A: N x N -> N as follows: A(m, n) ={2n, if m= 0; 0, if m≥1 and n= 0; 2, if m≥1 and n= 1; A(m-1, A(m, n-1)), if m≥1 and n≥2 Show that A(m, 2) = 4 whenever m≥1. Hint: Induct on m.

Define a function A: N x N -> N as follows: A(m, n) ={2n, if m= 0; 0, if m≥1 and n= 0; 2, if m≥1 and n= 1; A(m-1, A(m, n-1)), if m≥1 and n≥2 Show that A(1, n) = 2^n whenever n≥1. Hint: Induct on n.

(a) Is every total ordering a lattice? Why or why not?

(b) Let S={1,2,3}, and define the poset (P(S),⪯) by A⪯B if and only if A⊆B. Verify that this poset is a lattice. Is it a total ordering? (c) Using your work in part (b), is every lattice necessarily a total ordering?

Let P1={B0, B1, B2} be a partition of Z, where B0={3n|n ∈ Z}, B1={3n+ 1|n ∈ Z}, and B2={3n+ 2|n ∈ Z}. Describe the equivalence relation R1 corresponding to P1.

Let S be the set of ternary strings (i.e,. strings containing only the characters 0, 1,and 2), and let R be an equivalence relation on S. Suppose the collection of equivalence classes for R is P={Bi|i ∈ N}, where a typical representat…

Let R be an equivalence relation on Z, which has P={{±i}|i ∈ N} as its collection of equivalence classes. Describe the equivalence relation R.

If In= (-1/(5n),1/(5n)) where n≥1 is an integer and In represents an interval on the real number line, find U(n=1 to infinity) In and ∩(n=1 to infinity) In.

Let Bn={(x, y)|0≤x≤n and 0≤y≤n}, where n is a nonnegative integer. Find U(n=0 to infinity)Bn and ∩(n=0 to infinity)Bn.

Let C={A1, A2, ..., An} be a collection of finite sets that are pairwise disjoint. Further suppose that |Ai|=i. Compute |U(i=1 to n)Ai|, and write your answer in the simplest closed form possible.

a) Show that a subset of a countable set is countable.

If X,Y, and Z are sets and |X|=|Y| and |Y|=|Z|, show that |X|=|Z|. Note that we are not assuming that the given sets are finite.

Let A be a countable set, and B is another set. Assume further that there exists an onto function f:A->B. Is B necessarily countable? Provide a full justification for your answer.

Write following using summation n(n-1)+(n-2)++.....+1

Define a function fm: N x N ->N as follows: fm(n, k) =k if 0 ≤ n < m, and fm(n, k) =fm(n-m, k+1) otherwise. Describe in terms of a single well-known arithmetic operation what fm(n, 0) is computing.

Consider the statement: The art show was enjoyable but the room was hot. a) Use a variable to represent each basic statement in the given statement. b) Use the variables and logical operators to represent the statement in symbolic for…

Show that A=B A = {1,2,3} B ={n|n∈Z+ and n^2<10}

Let A = {2,3,4} , B = {6,8,10} , and C = {a,b} True or False? (i) 4R6 (ii) 4R8 (iii) (3,8) ∈ R (iv) (2,10) ∈ R (v) (4,12) ∈ R

There are 18 mathematics majors and 325 computer science majors at a college. a) In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major? b) In how many ways can …

Q 4: Find the Number of Mathematics students in a university taking at-least one of the language Mandarin, English and Japan Give the following data: 65 study Mandarin 45 study English 42 study Japanese 20 Study Mandarin and English …

Q. Construct the truth tables for (p & q here are not above statement) a) p V ~q b) ~p V ~q c) q -->p d) ~(p-->q)-->r

how propositional logic can be used in Boolean search? Give detail answer with an example of web page searching.

The generating function of the sequence {1,2,3...n..} is (1-z)².

6. Use truth tables to determine whether the argument form is valid. [5 Marks] a. There is an undeclared variable or there is a syntax error in the ﬁrst ﬁve lines. b. If there is a syntax error in the ﬁrst ﬁve lines, then there is a m…

∀x∃y(x + y = 2 ∧ 2x − y =1

A palindrome number is a number that is read similarly backwards. How many possible 5-digit palindromic numbers are there? A. 608 B. 648 C. 688 D. 728

A keyboard has its keys removed. If the last line of the letters were jumbled, how many ways can I rearrange it so that consecutive letters are beside each other? A.120 B. 144 C. 156 D.196

Five different colored flags can be hung in a row to make a coded signal. How many signals can be made if a signal consists of the display of one or more flags? A. 11 B. 13 C. 15 D. 17

Make negation of the following statements. I4) All students have taken a course in mathematics. 15) Some students are intelligent, but not hardworking

In an engineering class there are 15 students taking chemistry,20 students taking mathematics. Of these students,10 are taking both chemistry and mathematics.How many students are taking either chemistry or mathematics? Expert's answer

Alice decides to set up an RSA public key encryption using the two primes p= 31 and p= 41 and the encryption key e= 11.You must show all calculations, including MOD-calculations using the division algorithm! Bob decides to send the me…

1. Discuss two real world binary problems in two different fields using applications of Boolean Algebra.

Write the following sets in the roster/tabular form: a. G = {x : x ∈ N, 5 < x < 12} b. H = {x : x is a multiple of 3 and x < 21} c. I = {x : x is perfect cube 27 < x < 216} d. J = {x : x = 5n - 3,n ∈ W, and n < 3}

prove with inductions that n^2+n divisible by 2

Prove that (1 * 2) + (2 * 3) + (3 * 4) + (4 * 5) + ....+ n (n + 1) = n (n + 1) (n + 2) / 3 for n positive integers

In a survey, 2000 people were queried if they read India Today or Business Times. It was found 1200 read India Today, 900 read Business Times and 400 read both. How many read at least one magazine

Suppose that the statement p→ ¬q is false. Find all combinations of truth values of r and s for which (¬q→r)∧(¬p∨s) is true.

Let p and q be the propositions ”Swimming at the Sarıyer shore is allowed” and ”Sharks have been spotted near the shore”, respectively. Express each of these compound propositions as an English sentence. (a) ¬p ∨ q (b) p ⇒ ¬q (c) p ⇔ …

Determine the truth values of each proposition below. (a) 1 + 1 = 3 if and only if 2 + 2 = 3. (b) If 1 + 1 = 2 or 1 + 1 = 3, then 2 + 2 = 3 and 2 + 2 = 4. (c) If squirrels play badminton, then cats can’t fly.

If p is true, q, r, and s are false, find the truth value of the proposition. ¬(¬𝑝 ∨ 𝑞) ∨ ¬((𝑝 ∧ ¬𝑟) → ¬𝑠)

There are 7 groups in a picnic who has brought their own lunch box, and then the 7 lunch box are exchanged within those groups. Determine the number of ways that they can exchange the lunch box such that none of them can get their own.

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of a particular suit are chosen

There are 7 groups in a picnic who have brought their own lunch box, and then the 7-lunch boxes are exchanged within those groups. Determine the number of ways that they can exchange the lunch box such that none of them can get their …

Draw the Hasse diagram for (i) Pi = {1, 2, 3, 4, 12} and < is a relation such that x < y if x divides y.

Let P = {1,2,3,4} and R = {(1,1), (2,1), (1,2), (2,2), (3,2), (3,4), (4,3), (4,4)} which is a relation on P. Represent this relation as a directed graph. Check whether this relation is an equivalence relation or not.

For discrete structures there are n exams to check and there are k graders. To guarantee a high quality of grading, every exam may be checked by any number of graders (but always at least by one grader). This means that summed all tog…

1. Expand the following using the Binomial Theorem a. (x-3y)^3 b. (2a+b^-3)^4 c. (a+b+c)² 2. Solve the recurrence relation Pn = Pn-1 + n with the initial condition p_1 = 2 using iteration.

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