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# Discrete Math

### Matlab Code

** Please see the attached file for the complete problem description ** Please prepare a simple and basic MATLAB code for the 2 attached problems. M-file and word documents with comments will help me understand the codes better. Thank you. 3) Write a recursive function computing the sum of cubes of all integers between 1 an

### Proof about Ordering the Real Numbers

Given any two real numbers x < y, we can find a rational number q such that x < q< y. Hint: you can prove this by using the following statement: for any positive real number x > 0 there exists a positive integer N such that x > 1/N > 0

### Loan Amortization Schedule Comparison Project

Develop a loan amortization project using Excel spreadsheets. This can be a car loan or a home mortgage. You will develop two alternative loan schedules using realistic rates and repayment schedules and write up a comparison of your two amortization schedules. This involves two tables and two line graphs displaying the decrease

### Proof Equivalent Cauchy Sequences

1) show that if (a_n)^infinity evaluated at n=1, and (b_n)^infinity evaluated at n=1 are equivalent sequences of rationals, then (a_n) ^infinity evaluated at n=1 is a Cauchy sequence if and only if (b_n)^infinity evaluated at n=1 is a Cauchy sequence. 2) Let epsilon >0. Show that if (a_n)^infinity evaluated at n=1 and (b_n)^

### Antichains of a power set

Would someone be able to explain to me how we would find antichains and chains of a powerset? For example, if we had the power set of 4 - P() then how would we derive the antichains, and symmetric chains? What about for the power set of ? (where  is the set of numbers {1,2,3,4,5}. Is there are a general formul

### Proof about union and cardinalities

Please help with the following problem. Provide a step by step explanation. Show that given finite sets A_1, A_2,...,A_n, that are pairwise-disjoint, that is A_i intersection A_ j = empty set for all i not equal to j, then their union is a finite set and the cardinality of their union is the sum of the cardinalities of the s

### Proof showing the equality of integers

Please help with the following problem. Provide step by step. Show that equality of integers is an equivalence relation, that is show that equality of integers is reflexive, symmetric, and transitive. Recall two integers z=a--b, w=c--d, a, b, c, d belong to N (natural numbers) are equal if and only if a+d=b+c **where a--b

### Various Problems in Discrete Mathematics

Prove Each Directly. 1. The product of any two even integers is even. Prove by cases, where n is an arbitrary integer and Ixl denotes the absolute value of x. 2. [-x]=[x] (*Brackets are the x's is the absolute value symbol) Give a counterexample to disprove each statement, where P(x) denotes an arbitrary p

### Providing Proof about Integers and Rationals

Prove the cancellation law for integers: If a, b, c are integers such that ac=bc and c is non zero, then a=b. Hint for proof* use the cancellation law: Let a, b, c be natural numbers such that ac=bc and c is non zero, then a=b Use the trichotomy of integers.

### Proof about Integers and Rationals

Show that the definition of negation on the integers is well-defined in the sense that if (a----b)=(a'----b'), then -(a----b)= -(a'----b') (so equal integers have equal negations) where a----b is the space of all pairs equivalent to (a,b)

### Correcting Proofs in a False Statement

Not every error in a proof is a false statement. Sometimes every line is true, but the lines don't directly follow the previous lines. All that's needed to fix the error is more detail or explanation. Explain where the error is in the following proof and then fix it. Prove: If n^2 is not divisible by 3, then n is not di

### MATLAB to End Last Statement

I want this attached code to continue,using the last statement which is i have failed to close 'end'.I know Iam missing something,I want it edited and saved in word.the second how I think it shld look like,but its not working

### Prove that if a binary operation on an identity element

Please formally prove that if a binary operation * on a set X has an identity element e (element of) X , then it must be unique. Please submit response as a PDF or MS Word file. Infinite Thanks.

### Providing proof about Cardinality

Let A, B, C be sets. Show that the sets (A^B)^C and A^(BxC) have equal cardinality by constructing an explicit bijection between the two sets. Conclude that (a^b)^c=a^bc for any natural numbers a, b, c. Use a similar argument to also conclude a^b x a^c= a^b+c

### Examining Proof about Cardinality

Let A and B be sets. Show that A x B and B x A have equal cardinality by constructing an explicit bijection between the two sets. Then use the following proposition to prove that multiplication is commutative. (Let n, m be natural numbers. Then nxm=mxn) Proposition: Cardinal arithmetic a) Let X be a finite set, and let x b

### Counting Methods Roulette Wheel Diagrams

Part I: Set Theory Look up a roulette wheel diagram. The following sets are defined: A = the set of red numbers B = the set of black numbers C = the set of green numbers D = the set of even numbers E = the set of odd numbers F = {1,2,3,4,5,6,7,8,9,10,11,12} From these, determine each of the following: A?B A?D B?C

### Proof about Set Theory

Let A, B, C be sets and let X be a set containing A, B, C as subsets. Prove that A intersects (B union C)= (A intersects B) union (A intersects C) A union (X relative complement of A)=X and A intersects (X relative complement of A)=empty set X relative complement (A intersects B)=(X relative compliment of A) union (X re

### Euclidean Algorithm

Prove the Euclidean Algorithm: Let n be a natural number, and let q be a positive number. Then there exist natural numbers m, r such that 0 < or = r < q and n=mq+r Show that m and r exist and are also unique. **Fix q and induct on n, and note that you can only use rules of addition

### Mathematical Reasoning: The Sum of the Divisors Function

Prepare written answers to the following assignments: Preview Activity 3 (The Sum of the Divisors Function) Let s be the function that associates with each natural number the sum of its distinct natural number factors. For example, s (6) = 1 + 2 + 3 + 6 = 12. 1. Calculate s (k) for each natural number k from 1 thr

### Discrete math Subsets Contained

Hello, I have another discrete problem I need help on. It says: How many subsets contain 1 or 2 or 3 in the set {1,2,...,20}? So my teacher told me that {1} is a subset, {1,3,4,5,19} would be a subset (i just chose that randomly), {2} would be a subset, {2,5,6,7,20} would be a subset (again, i just chose that at random

### How many solutions do the equation have

Hello, I have a discrete math problem. It states how many solutions (X_1,...,X_6) in the Natural numbers have the equation X_1+ ... + X_6 = 10 So the way my teacher explained it was (10,0,0,0,0,0) would be one choice, (0,10,0,0,0,0) would be another, so basically it would look like this... (10,0,0,0,0,0) (0,10,0,0,0,0

### Discrete Math: Proving a theorem

(a) Proof. Let f be onto. Consider any C is a subset of Y. Let y E f(f^-1(C)). Then y=f(x) for some x E F^-1(C). But the fact that x E f^-1(C) implies that f(x) E C. Moreover, f(x)=y. Therefore y E C. Thus we have proved that f(f^-1(C)) is a subset of C. For the converse, consider any c E C. Since f is onto, there exists a

### SAT and ACT Scores

Bob compares his SAT Verbal score of 400 to Marge's ACT Verbal score of 20. "I whupped ya," he exclaims. "My score is 20 times your score!" Although Bob's multiplication is good, his logic is faulty. Exxplain why?

### Ordered Pairs and Hasse Diagram Help

Let A = {a, b, c} and let R be the relation defined on A defined by the following matrix: M=R = [1,0,0; 1,1,0; 0,1,1 (a) Describe R by listing the ordered pairs in R and draw the digraph of this relation. (b) Is this relation a partial order? Explain. If this relation is a partial order, draw its Hasse diagram.

### Listing the distinct equivalence classes

Let X={1,2,3,4,5}, Y={1,2}. Define relation R on g(x) by ARB iff AY =BY *Note: g(x) is the power set of x and R is a equivalence relation (no need to prove this)* a) C={2,3}. List the elements of [C], the equivalence class containing C. b) How many distinct equivalence classes are there? c) Suppose X={1,2,...,n

### Discrete Math and Integral Solutions

Please see attached. Thanks

### Deriving the quation of line

The percentage of homes with digital TV services stood at 5% at the beginning of 1990 (t=0) and was projected to grow linearly so that, at the beginning of 2003 (t=4), the percentage of such homes was 25%. a) Derive an equation of the line passing through the points A(0,5) and B(4,25). b) Using the equation found in part

### Apply the Gram-Schmidt Algorithm.

S=span{x_1, x_2, x_3}, where x_1=(1,1,1,-1)^T, x_2=(2,-1,-1,1)^T, and x_3=(-1,2,2,1)^T. 1) use Gram-Schmidt algorithm to the sequence {x_1,x_2, x_3} to find an orthonormal basis of S 2) use the result above to find the QR factorization of the matrix A=(x_1l x_2l x_3).

### Prove that a given binary operation on a set is associative, but the set with this operation is not a group.

On the set Z of integer numbers we are given a binary operation o as follows: n o m = n + m + nm, for any n, m E Z. Prove that this operation is associative, however the pair (Z, o) is not a group.

### Solving a Strange Summation

First note that x = Floor(x) + {x}, where {} is the remainder. So if x= 2.74, then the floor is 2 + the remainder is .74 . f(x)= { 0 if x its a perfect square (like 1,4,9,...) { Floor (1/ {sqrt(x)}) otherwise So for the otherwise part, say x=2, you have 1/the remainder of sqrt(2), which is 1/.414 = 2.4