Solved: Qu 01: Model the following situations as (possibly weighted, possibly directed) graphs. Draw each graph, and give the corresponding adjacency matrices. (i) Rokey and Jane are friends. Rokey is also friends with Kevin and Ferdy. Jane ,Kevin and kanny are all friends of each other.

Define a bijective function. Explain with reasons whether the following functions are bijective or not. Find also the inverse of each of the functions. i. f(x) = 4x+2, A=set of real numbers ii. f(x) = …

Let f and g be functions from the positive real numbers to positive real numbers defined by f(x) = [2x] g(x) = x2.2 Calculate f o g and g o f.

Define partial order and total order of relations

In the given picture are three men: Neil Armstrong, Michael Collins and Buzz Aldrin. They were on the Apollo 11 that set the first man on the moon in 1969. Neil Armstrong was the first man walking on the moon. Which is an example of a…

Show that the power set of S={a,b,c} is a poset under set inclusion

If the truth value of ( p ➡️ q) v negation r is false, then what is the truth value negation q ↔️r ?

For the following, find if there are any errors in the methods of proof given below. List out these errors and write how you would prove/disprove the statements given below. (a) Statement: If n is an integer and n^2 is divisible by 4,…

[P ۷ (P ۸ Q)] → R

For divisibility relation on the set {1,2,3,6,8,12,24,36}, draw Hasse diagram. Then find minimal, maximal, greatest and least elements. Then give the topological sort using the divisibility relation

For divisibility relation on the set {1,3,6,9,12,5,25,125}, draw Hasse diagram. Then find minimal, maximal, greatest and least elements. Then give the topological sort using the divisibility relation

Solve the recurrence relation an=6 an-2-12 an-2 + 8 an-2 + n2 2n

using havels’s –hakim verify if the degree sequence (3,3,2,2,1,1) is a graphics,if so then draw the graph.

find the transitive closure of (1 3) (5 5) (1 6) (3 3) (5 6)

Determine whether the following relations are injective and/or subjective function. Find universe of the functions if they exist. i. A= v,w,x,y,z, B=1,2,3,4,5 R= (v,z),(w,1), (x,3),(y,5)

Determine whether the following relations are injective and/or subjective function. Find universe of the functions if they exist. A = 1,2,3,4,5 B=1,2,3,4,5 R = (1,2),(2,3),(3,4),(4,5),(5,1)

If a function is defined as f(x,n) mod n. Determine the i. Domain of f ii. Range of f iii. G(g(g(g(7)))) if g (n) = f(209, n).

Let L be a lattice. Then prove that a Ù b=a if and only if a v b=b.

Define the dual of a statement in a lattice L. why does the principle apply to L.

Let X = {1,2,3,4,5,6,7} and R = {x,y/x–y is divisible by 3} in x. Show that R is an equivalence relation.

Let A = {1,2,3,4} and let R = {(1,1), (1,2),(2,1),(2,2),(3,4),(4,3), (3,3), (4,4)} be an equivalence relation on R. Determine A/R.

Draw the Hasse diagram of lattices, (L1,<) and (L2,<) where L1 = {1, 2, 3, 4, 6, 12} and L2 = {2, 3, 6, 12, 24} and a < b if and only if a divides b.

What is a partial order relation? Let S = { x,y,z} and consider the power set P(S) with relation R given by set inclusion. Is R a partial order

Define a lattice. Explain its properties.

Show that p ⋁ (q ⋀ r) and (p ⋁ r) ∧ (p ⋁ r) are logically equivalent. This is the distributive law of disjunction over conjunction.

Let P be the power set of {a, b, c}. Draw the diagram of the partial order induced on P by the lattice (P,(,().

Draw the Hasse diagram of lattices, (L1,<) and (L2,<) where L1 = {1, 2, 3, 4, 6, 12} and L2 = {2, 3, 6, 12, 24} and a < b if and only if a divides b.

Let X = {a, b, c} defined by f : X (X such that f = {(a, b), (b, a), (c,c)}. Find the values of f^1, f^2 and f^4.

Determine whether the following relations are injective and/or subjective function. Find universe of the functions if they exist. i. A= v,w,x,y,z, B=1,2,3,4,5 R= (v,z),(w,1), (x,3),(y,5) ii. A = 1,2,3,4,5 B=1,2,3,4,5 R = (1,2),(2…

If a function is defined as f(x,n) mod n. Determine the i. Domain of f ii. Range of f iii. G(g(g(g(7)))) if g (n) = f(209, n).

StatethevalueofxafterthestatementifP(x)thenx:=1 b)x=1. a)x=0. isexecuted,whereP(x)isthestatement“x>1,”ifthe valueofxwhenthisstatementisreachedis c)x=2.

a={x/x is the letters of the word mississippi}

Let X = {a, b, c} defined by f : X ®X such that f = {(a, b), (b, a), (c,c)}. Find the values of f–1, f2 and f4.

Determine whether the following relations are injective and/or subjective function. Find universe of the functions if they exist. i. A= v,w,x,y,z, B=1,2,3,4,5 R= (v,z),(w,1), (x,3),(y,5) ii. A = 1,2,3,4,5 B=1,2,3,4,5 …

If a function is defined as f(x,n) mod n. Determine the i. Domain of f ii. Range of f iii. G(g(g(g(7)))) if g (n) = f(209, n).

Determine whether the following relations are injective and/or subjective function. Find universe of the functions if they exist. i. A= v,w,x,y,z, B=1,2,3,4,5 R= (v,z),(w,1), (x,3),(y,5) ii. A = 1,2,3,4,5 B=1,2,3,4,5 R = (1,2),(2…

If a function is defined as f(x,n) mod n. Determine the i. Domain of f ii. Range of f iii. G(g(g(g(7)))) if g (n) = f(209, n).

Identify the error or errors in this argument that supposedly shows that if ∃xP (x) ∧ ∃xQ(x) is true then ∃x(P (x) ∧ Q(x)) is true. ∧

A) Let P be the propositions Roses are Red and Q be the propositions Violets are Blue. Express each of the following propositions as logical expressions: (I) If roses are not red, then violets are not blue. (II) Roses are Red or Vio…

By constructing truth tables, decide which of the following are tautologies. (I) ~(P ^ ~P) (II) P implies ~P (III) (P ^ (p implies q)) implies q B) show that (P implies Q) implies R is logically equivalent to (~ P implies R) ^ (Q …

Give a proof of the theorem, "if n is odd, then n squared is odd".

For the following functions, giv a tabular representation of their Boolean expressions; (I) F(X,y,z) = -(X,z+y) (II) F(X,y,z) = ((xz)) (-y) (III) what can you say about (I) and (II) above?

Prove by induction that 1+2+3+.....+n = n(n+1)÷2

In the following argument, determine the validity or otherwise of the statement: "If the BIOS test runs fine, the CPU and motherboard must be OK. If the CPU and the motherboard and memory are all OK, then there must be a flaw in …

Define the following with an example (I) tautology (I) proposition

Let A = {a, b, c}, B = {x, y} and C = {0, 1}. Find a. C x B x A b. B x B x B

Let p and q be the propositions: P: I will do every exercise In this book. Q: I will get an 'A' in this course. Write the following propositions using p and q and logical connectives: (I) For me to get an 'A' in this course, it is n…

Let p,q and r be the following proposition P:jun jun has a date with petra Q:juan is sleeping R:lance is eating 1.p v q 2.q V (～r) 3.p V (q v r)

Use set builder notation to give a description of this set {0,3,6,9,12}

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

Translate the following sentences into propositional logic) If you do not do homework by yourself or copy from other places, then you will get 0 in homework.

Prove the following equivalences by the logical derivation: (b) (p ∧ q) → r ≡ (p → r) ∨ (q → r)

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. a) ∃x(x2 = −1) b) ∀x(x2 + 2 ≥ 1)

 Let P(x), Q(x), and R(x) be the statements “x is a lion,” “x is fierce,” and “x drinks coffee,” respectively. Assuming that the domain consists of all creatures, express the statements in the argument using quantifiers and P(x), Q(x…

{xlx is an integer such that x

Use a Venn diagram to illustrate the set of all months of the year whose names do not contain the letter R in the set of all months of the year.

Use proof by contradiction to show that “if m and n are odd integers, then m + n is even.”

By using principle of mathematical induction prove 2^n > n^2 if n is an integer greater than 4.

Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if a) x + y = 0. b) x = ±y. c) x - y is a rational number.

Determine if the statement is TRUE or FALSE. Justify your answer. All numbers under discussion are integers. 1.For each m ≥ 1 and n ≥ 1, if mn is a multiple of 4, then m or n is a multiple of 4. 2. For each m ≥ 1 and n ≥ 1, if m…

What is sets??

Prove: There is no positive integer n such that n^2+n^3=100.

Prove: If ab is even then a or b is even.

For n∈Z, prove n^2 is odd if and only if n is odd.

determine whether these functions are bijections f:Q to R x2+1/x

(i). In MA161 class, the lecturer gave the following proposition. Let A (x, y) be “ y is greater than or equal to x ”. The domain for the proposition is the set of nonnegative integers. The students were tasked to determine what are t…

Draw a Venn diagram that represents V, the set of vowels in the English alphabet, where set of vowel is {a,e,i,o,u}

Let A = {0, 2, 4, 6, 8}, B = {0, 1, 2, 3, 4}, and C = {0, 3, 6, 9}. What are A ∪ B ∪ C and A ∩ B ∩ C?

How many ways to arrange the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 in two(2) rows of six (6) columns so that (1) the integers increase in value as each row is read, from left to right and (2) in any column the smaller integ…

I. Use quantifiers to express the following statements. 1. Every student enrolls in a math class. 2. There is a student in a class who drops a math course. 3. Every student in this class wears an ID. 4. At least one student in thi…

Let P(x) be the statement “x = x²." If the domain con- sists of the integers, what are these truth values? c) P(2) f) VxP(x) a) P(0) b) P(1) d) P(-1) e) 3x P(x)

Draw these graphs. a) K7 b) K1,8 c) K4,4 d) C7 e) W7 f ) Q4

Can a simple graph exist with 15 vertices each of degree five?

Write these propositions in symbols using p, q, and r and logical connectives. You get a 95 in math, but you do not do every assignment in the class

A. Use quantifiers to express the following statements. 1. Every student enrolls in a math class. 2. There is a student in a class who drops a math course. 3. Every student in this class wears an ID. 4. At least one …

Let A = {2, 4} and B = {6, 8, 10} and define relations R and S from A to B as follows: for all (x,y) EA x B. XRyoxly for all (x,y) EA x B x Syy-4 = x State explicitly which ordered pairs are in A x B. R, S, RUS and Ros. And draw the C…

R1 = {(4,5)} R2 = {(1,5), (1,6), (1.7), (1,8), (2,5), (2,6), (2,7), (2,8), (3,5), (3,6), (3,7), (3,8), (4,5), (4,6), (4,7), (4,8)} Evaluate R1 ◦ R2 and R2 ◦ R1

Use a Venn diagram to illustrate the subset of odd integers in the set of all positive integers not exceeding 10.

If (S,*) is a.semigroupand x € s show that (S,∆) is a semigroup if a∆b =a*x*b

Verify the validity of the argument: All lions are fierce. Some lions do not drink coffee. Hence some fierce creatures do not drink coffee.” (Lewis Carrol)

LetA={0,2,4,6,8},B={0,1,2,3,4},andC={0,3,6,9}.Find the following: A∪B∪C A∩B∩C (A∪B∪C)𝐶 4.(A∩B∩C)𝐶 5.(A∪B)∩C)𝐶

mathematical notations for The set of all even numbers.

Use set builder notation to give a description of each of these sets. a. {0, 2, 4, 6, 8,10,12,14,16} b. {−3,−2,−1, 0, 1, 2, 3}

how many ways to arrange the integers 1,2,3,4,5,6,7,8,9,10,11,12 in two rows og six columns so that 1 the integers increase in value as each row is read, from left to right and 2 in any column the smaller integer is on top.

Find out if the following sets are Countable, Uncountable, Finite or if it cannot be determined. Give the reasoning behind your answer for each. (a) Subset of a countable set (b) integers divisible by 5 but not by 7 (c) (3, 5) (…

(a) Find the solution to an = an-1 + 2n + 3 with the initial conditions a0= 4. (b) Consider the recurrence an = an-1 + 2an-2 + 2n - 9 show that this recurrence is solved by: i. an = 2 - n ii. an = 2 - n + b * 2n for any real b.

Given two sets A and B, for each of the following statements, what can you conclude about the sets? For example: consider the statement A−B=∅, this could be possible if - Scenario - 1: if A=B then A−B=∅ Scenario - 2: Since A−B=A−(A…

Give an example of two uncountable sets A and B with a nonempty intersection, such that A−B is (a) Finite (b) Countably infinite (c) Uncountably infinite

Consider the following functions and determine if they are bijective. [A function is said to be bijective or bijection, if a function f: A→B is both one-to-one and onto.] (a) f: Z × Z→Z, f(n, m) = n2 + m2 (b) f: R→R, f(x) = x3 − 3 …

Let X=Y = R and Z = set of all integers. Let f:Y-Z defined by fix) = [x], and g:X-Y be defined by g(x) = (5x-3)/2 Find: 1. f(-5/2) 2. f(n) 3. g(1) 4. fo g (x)

Let S = {Barnsley, Manchester United, Southend, Sheffield United, Liverpool, Maroka Swallows, Witbank Aces, Royal Tigers, Dundee United, Lyon} be a universal set, A = {Southend, Liverpool, Maroka Swallows, Royal Tigers}, and B = {Barn…

2. 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}

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 ,3take chocolate, tea,and cocoa ,20 take cocoa only ,2 take tea,coffee an…

In the given picture are three men: Neil Armstrong, Michael Collins and Buzz Aldrin. They were on the Apollo 11 that set the first man on the moon in 1969. Neil Armstrong was the first man walking on the moon. Which is an example of…

Determine the validity of the following argument. “To pass the Discrete Mathematics, it is necessary to pass both the course work and the final examination. Either John will have to work hard in Discrete Mathematics or he will fai…

SOLUTION a) ∀x(x2 ( x) solution x=2,y=1,x\not= \frac{1}1x=2,y=1,x  = 1 1 b) ∀x(x2 ( 2) solution x=1,y=20, \\20^2-1>100x=1,y=20, 20 2 −1>100 c) ∀x(|x| > 0 solution x=8,y=4,\\8^2=4^3x=8,y=4…

Qu 01: Model the following situations as (possibly weighted, possibly directed) graphs. Draw each graph, and give the corresponding adjacency matrices. (i) Rokey and Jane are friends. Rokey is also friends with Kevin and Ferdy. Jane ,Kevin and kanny are all friends of each other.

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Question ID: mtid-5-stid-8-sqid-2897-qpid-1596