Solved: Draw the directed graphs representing each of the relations on {1, 2, 3, 4} a) {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} b) {(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)} c) {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)} d) {(2, 4), (3, 1), (3, 2), (3, 4)}

Instruction: Write only the letter of your answer. 1. Suppose p is the statement 'You need a credit card' and q is the statement 'I have a nickel.' Select the correct statement corresponding to the symbols ┐(p ∨ q). A. You don't need …

4. Suppose p is the statement 'I play softball' and q is the statement 'The moon is 250,000 miles from Earth.' Select the correct statement corresponding to the symbols ┐p ∧ q. A. I don't play softball and the moon is 250,000 miles fr…

7-13: Find the Truth Value (show your solution) 7. Suppose p is false, q is false, s is true. Find the truth value of (s ∨ p) ∧ (q∧ ┐s) 8. Suppose p is true, q is true, r is false, s is false. Find the truth value of (s ∨ p) ∧ (┐r ∨ ┐…

6. Select the statement that is the negation of “You wear matching socks to the interview or you don’t get hired.” A. You don’t wear matching socks to the interview or you get hired. B. You don’t wear matching socks to the interview a…

Explain graph isomorphism with proper diagrams.

The recursive definition of a function X is given as: f(0)=5 and f(n)=f(n-2)+5 Now, find out the value of f(14) using the above function.

Consider the relation on the set of integer R={(a, b) /a=b+1} check whether it is equivalence relation

Among the integer 1 to 300,find how many are not divisible by 3,nor by 5 also find how many are divisible by 3but not by 7

The recursive definition of a function X is given as: f(0)=5 and f(n)=f(n-2)+5 Now, find out the value of f(14) using the above function.

How many bits would be required to represent decimal numbers from -32,768 to +32,767?

Each of the following numbers represents a signed decimal number in the 2’s complement system. Determine the decimal value in each case  10011001  11101  01111011

DISCUSS AND ILLUSTRATE WITH EXAMPLES. 15 POINTS EACH QUESTION. B. Universal Relations C. Identity Relations D. Inverse Relations E. Reflexive Relations F. Symmetric Relations G. Transitive Relations

For the function f defined by f(n) =n2+ 1/n+ 1 for n∈N, show that f(n)∈Θ(n). Use

If n is any even integer and m is any odd integer, then (n + 2)2 - (m - 1)2 is even.

Suppose a ϵ Z. Prove by contradiction that if a2 -2a + 7 is even, then a is even.

Use proof by contraposition to show that if x + y ≥ 2, where x and y are real numbers, then x ≥ 1 or y ≥ 1.

Suppose a ϵ Z. Prove by contradiction that if a2 -2a + 7 is even, then a is even.

Suppose that in the world every pair of people either (a) likes one another, (b) dislikes one another, or (c) is indifferent toward one another. Prove that in any gathering of 17 people, there is a group of three people all of whom sa…

In a a class of 80 students 53 study Art 60 study Biology 36 study art and Biology 34 study art and Chemistry 6 study Biology only and 18 study biology but not chemistry. How many students study art only and how many students study ch…

. The crossing number of a graph G, written ν(G), is the fewest number of nonendpoint edge-crossings that occur over all possible drawings of G in the plane. We assume that no edge crosses itself, and that edge crossings occur only at…

WHAT IS THE CARDINALITY OF {a,{a},{a,{a}}}

Use Karnaugh map to minimize the sum of product expansion xy'z+ xy'z'+x'yz+x'y'z+x'y'z'

Determine whether f is a function from the integers to the set of all real numbers. Enter "Y" for yes and "N" for no. 1. f(n)=±n 2. f(n)=1/(n^2−16) 3. f(n)=√n^2+6 4. f(n)=1/(n^2+9)

Find the following values. (a) ⌊1.1⌋ (b) ⌈1.1⌉ (c) ⌊−0.1⌋ (d) ⌈−0.1⌉ (e) ⌈2.99⌉ (f) ⌈−2.99⌉ (g) ⌊1/2+⌈1/2⌉⌋ (h) ⌈⌊1/2⌋+⌈1/2⌉+1/2⌉

Determine if each of the following functions from {a,b,c,d} to itself is one-to-one and/or onto. Check ALL correct answers. (a) f(a)=d,f(b)=a,f(c)=c,f(d)=b A. onto. B. neither one-to-one nor onto. C. one-to-one. f(a)=b,f(b)=a,f(c)=c…

Given that f(x)=8x^2+7 and g(x)=4x+4 are functions from R to R, find (a) f∘g. (b) g∘f.

(a) How many bit strings of length 8 are there? (b) How many bit strings of length 8 or less are there? (Count the empty string of length zero also.) (c) How many strings of 6 lower case English letters are there that have the letter …

Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties: (a) have distinct digits. (b) are divisible by 5 and by 7. (c) are even. (d) …

How many strings of four decimal digits (Note there are 10 possible digits and a string can be of the form 0014 etc., i.e., can start with zeros.) (a) do not contain the same digit twice? (b) begin and end with a 1?

How many strings of five uppercase English letters are there (a) that start or end with the letters BO (in the order), if letters can be repeated? (inclusive or) (b) that start with the letters BO (in that order), if letters can be re…

Solve the following two " union " type questions: (a) How many bit strings of length 8 either begin with 2 0s or end with 1 1s? (inclusive or) (b) Every student in a discrete math class is either a computer science or a math…

bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. (a) How many balls must she select (minimum) to be sure of having at least three blue balls? (b) How many balls must she select (mi…

This question concerns bit strings of length six. These bit strings can be divided up into four types depending on their initial and terminal bit. Thus the types are: 0XXXX0, 0XXXX1, 1XXXX0, 1XXXX1. How many bit strings of length six …

Find the value of each of the following quantities: C(8,4)= C(12,7)= C(12,8)= C(11,1)= C(10,9)= C(8,6)=

How many bit strings of length 14 have: (a) Exactly three 0s? (b) The same number of 0s as 1s? (d) At least three 1s?

15 players for a softball team show up for a game: (a) How many ways are there to choose 10 players to take the field? 10!/(5!) (b) How many ways are there to assign the 10 positions by selecting players from the 15 people who show…

Suppose that a department contains 8 men and 20 women. How many ways are there to form a committee with 6 members if it must have strictly more women than men?

How many ways are there to select 9 countries in the United Nations to serve on a council if 1 is selected from a block of 50, 2 are selected from a block of 67 and 6 are selected from the remaining 72 countries?

Find the coefficient of x^5 in (1+x)^11.

What is the coefficient of x^3 y^13 in the expansion of (−1x+2y)^16?

Out of 300 students taking discrete mathematics, 60 take coffee, 27 take cocoa 36 take tea, 17 take tea only, 47 take chocolate only, 7 take chocolate and cocoa ,3 take chocolate, tea and cocoa ,20 take cocoa only ,2 take tea, coffee …

1) 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 (includ…

Prove or disprove: every transitive relation on a set X with more 2 points is reflexive

A. Give a brief definition of the following: 1. Graph 2. Degree of a vertex 3. Isomorphic graphs B. Give 4 types of graphs and give a brief description (you may describe in words or just draw a sample graph).

Eulerian and Hamiltonian Graphs, Weighted Graphs A. Define the following: 1. Walk 2. Path, Trail 3. Cycle, Circuit B. What is Eulerian Graph? C. What is Hamiltonian Graph? D. Describe how to solve the Konigsberg Problem.

A. Give a summary of the Greedy Algorithm. B. Give a summary of the Edge-Picking Algorithm. C. What is a graph coloring, and how is it applied?

Construct the Hasse diagram for ({1, 2, 3}, <=)

Let R be the relation on the set A = {a, b, c, d, e, f} and R = {(a,c), (b,d), (c,a), (c,e), (d,b), (d,f), (e,c), (f,d)} Find the transitive closure pf R using Warshall’s algorithm.

Given set S = {1, 2, 3, 4, 5, 6} and a partition of S, A1 = {1, 2, 3} A2 = {4, 5} A3 = {6} Find the ordered pairs that make up the equivalence relation R produced by that partition

Consider following relation on set (1, 2, 3, 4, 5, 6} R= {(i,j) | |i-j|=2} Is R transitive? Is R reflexive? Is R symmetric?

How many bit strings of length 8 either start with 1 bit or end with 2 bits 0

Out of 5 male and 6 female a committee of 5 is to be formed. Find the number of ways in which it can be formed so that among the person chosen in the committee there are: i) exactly 3 males and 2 females ii) at least 2 males and one f…

Let F(x, y) be the statement "x can fool y," where the domain consists of all people in the world. Use quantifiers to express each of these statements.

Define the relation of d on A by xdy if x is contained within y. For example, 01d101. Draw a digraph for this relation.

In a group of 25 people, there is a ____ chance that at least two of them has shaken hands with the same number of people.

For every integer n≥1,(n3+11n)⋅(8n−14n+27) is divisible by 6 7 42 84

If A={1,2,3,6,12,18,24,30,36,150} and the partial ordering relation is divides then draw the hasse diagram

Let A= {1,2,3,4,5,} and R be a relation on A, such that R = [(1,4), (2,2), (3,4), (3,5), (4,1), (5,2), (5,5)]. Use warshall’s Algorithm to find the matrix of transitive closure of R

Let A= {1,2,3,4,5,6} and R be a relation on A, such that R = [(x, y): |x-y| =2]. Use warshall’s Algorithm to find the matrix of transitive closure of R. and draw its digraph

Let A= {1,2,3,4,6,9,12} let a R b if a is divided by b. Show that R is POSET, Draw Hasse Diagram. Prove or disprove if it is a Lattice.

Draw the graph and its equivalent Hasse diagram for divisibility on the set {1, 2, 3, 6, 12, 24, 36, 48}

a) Let A = (1,2,3,4) and R = ((1 ,2),(2,4),( 1,3 ),(3 ,2)}. Find the transitive closure of R by Warshall’s algorithm. b) Let A = {a,b,c}. show that (P(A),c ) is a poset and draw its Hasse diagram. c) Define Chains and Antichains.

Let R be a binary relation on the set of all positive integers such that R = { (a,b) | a- b is an odd positive integer } Is R reflexive, symmetric, antisymmetric, transitive? Is R an equivalence relation? A partial ordering relation

Function f, g, h are defined on a set X = (1, 2, 3) as f = ((1, 2), (2, 3), (3, 1)} g = {(1,2),(2,1),(3,1)} h = ((1,1), (2,2),(3,1)} i) Find fog, gof. Are they equal? ii) Find fogoh and fohog. Define partial function

In a class has 124 students, of these students 47 are involved in math club, 25 involved in science club and 36 involved in English club so how many are involved in both physics and English club?

Calculate the number of bit strings of length 5 or less.

(2) How many different license plates can be made consisting of 10 characters if the first 5 characters consist of uppercase English letters, while the final 5 characters are digits?

Compute the number of functions f from the set {0,1,2,...; n} (where n is a positive integer), to the set {0,1}.

Calculate the number of one-to-one functions there are from a set with 6 elements to sets with the following numbers of elements: (a) 5 (b) 6 (c) 7 (d) 8

Use the Pigeonhole Principle to show that if there are 31 students in a class, then at least two have first names that begin with the same letter.

Let d be a positive integer. Show that among any group of d+ 1 (not necessarily consecutive) integers there are two with exactly the same remainder when they are divided by d.

Assume that a class consists only of students residing the 50 US states. Find the smallest number of students that must be enrolled in a class to guarantee that there are at least 5 students from the same state

8) How many different 6-permutations are there of elements in the set {a, b, c, d, e, f, g, h}?

Compute the number of bit strings of length 14 that contain... (a) exactly four 1s (b) at most four 1s (c) at least four 1s (d) an equal number of 0s and 1s.

Let S be a finite set, with |S|= 200. Find the number of subsets of S containing more than 2 elements

1) Suppose a group contains 7 men and 7 women. How many ways are there to arrange these people in a row if the men and women alternate?

12) Suppose that a room contains 10 cats and 15 dogs. How many ways are there to form a committee consisting of 6 of these animals if it must have more dogs than cats?

Find the number of terms in the complete expansion of (x+y)^150 after like terms are collected together.

Find the coefficient of x^7 y^9 in the expansion of (4x+ 5y)^16.

Let S be a finite nonempty set. Show that the number of subsets of S of even cardinality equals the number of subsets of S of odd cardinality.

Use Mathematical Induction to prove the Binomial Theorem.

Let S={a, b, c, d, e}. Find the number of ways to select 7 elements from S when repetition is allowed. The order in which the elements are chosen does not matter

How many solutions are there to the equation x1+x2+x3+x4+x5+x6= 25 where the xi(for i= 1,2,3,4,5,6) are non-negative integers?

Compute the number of distinct arrangements of the characters ABBCCCDDDDEEEEE

How many bit strings can be formed by using seven 1s and nine 0s?

Which is the following statements is True? Every path is a trail but every walk is not a trail Every walk is a trail but every path is not a trail. Both paths and walks are trails. Neither a path nor a walk is a trail 1 point L…

Q.Which of the following can be a degree sequence of a simple graph? {4,4,3,2,1} {4,3,2,1} {4,4,3,2} {4,2,2,2,2} Q.If G is a connected simple graph and e,f and g are edges of G, then which of the following is/are true: There …

Width of a tree is defined as the largest distance(number of edges in the shortest path) between any two vertices. What is the smallest possible width of the tree with 100 vertices? 1 2 99 101 1 point How many complete graphs w…

Find a counter example to the following universally quantified statements, where the domain for all variables consist of all integers. ∀x, x^2, <=x^3.

given: f(n) = 3f(n/2) + 5n^3 where n=2k and f(1) = 2 find f(n) and a big -0 estimate of f(n)

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Question ID: mtid-5-stid-8-sqid-668-qpid-553