Category: Discrete Math

Professional Connections With Discrete Mathematics
Discrete mathematics can be viewed as a set, or collection, of a number of concepts spanning algorithms, logic, matrix operations, and set theory. Basic applications of these concepts can seem fairly clear as you study them, while the further translation, or transfer, in applicability to the real world or to your profession can be harder to grasp without focused research and reflection. This discussion is an opportunity for you to market yourself to a prospective employer by presenting how the concepts and skills you have learned in this class may be applied to your current or future profession.You will write a college-level learning narrative in which you write specifically about how the skills and concepts you have learned in class are useful to a current or future employer. Choose two specific concepts or skills that you have studied in this course and research how they may be applied to your current or future profession. You will highlight specific examples related to each concept as evidence of your learning and express key ideas related to each concept which you discover are important in your field based on evidence from your research.
As you draft the body paragraphs of your learning narrative, address the following questions for each of your two concepts:
How is the concept applicable to the field you would like to pursue?
From your research, what sources discuss this application, and how do they characterize applying the concept in the profession?
Share at least one specific example of a problem or skill you learned to show your prospective employer your understanding of this concept.
Share at least two key ideas that you need to highlight during an interview.
Is knowledge of the concept reflected in the required skills for a job advertisement you might consider? If so, describe the job and what part of the job advertisement reflects the need to possess knowledge of this concept.
The learning narrative should include a title page and reference page, as well as at least 1-page double-spaced in 12-point Times New Roman font. Every piece of writing should have an introduction, body, and conclusion. You are expected to use outside resources for this essay. Cite at least two sources in proper APA format on a reference page. You must also use appropriate APA in-text citations throughout your narrative.
Your writing should include a highly developed purpose and viewpoint; it should also be written in Standard English and demonstrate college-level content, organization, style, grammar, and mechanics. There should be no evidence of plagiarism. If you are unsure about what constitutes plagiarism, please review the Plagiarism Policy.
Talk about algorithms and graphs.

Teacher instructions:
Pleace complete it on the printed or copied sheet of paper, and then submit those two pages only. Be sure to show the work required for each problem. Thanks

The live assignment starts at 4 PM EST
The time limit is 1 hour 50 minutes. All of the questions will be displayed at the start of the live assignment
Work needs to be shown on paper for each question

The live assignment starts at 4 PM EST
The time limit is 1 hour 50 minutes
Work needs to be shown on paper for each question

Hey how you doing? I have 11questions that we have to finish in 1:30 min. I’ll send you pictures from here or your mobile number that’s not a problem. It’s about function, implicit differentiation, draw a table

Here is the prompt, there are five steps in total:
Consider a real-world situation that involves relationships that can be modelled with a graph. A
graph consists of a discrete number of vertices and the edges that connect them. When
brainstorming the situation you would like to model with a graph, consider reviewing the
examples that have been presented in your module readings and practice exercises for ideas.
1) Consider a situation in your personal or professional world that involves relationships that can
be modelled with a graph. Describe this situation, in at least one well composed paragraph,
sharing:
● A brief description of the situation modeled,
● What each vertex represents, and
● What each edge represents.
2) Draw a graph using a drawing program of your choice and include it in your post. The
following must be present in your graph:
● Five to ten vertices, each clearly labeled with a single capital letter (A, B, C, D, E …)
● At least two vertices of degree 3 or more (the degree of a vertex is the count of how
many edges are attached to that vertex).
● At least one circuit.
● Be a connected graph.
3) Consider your graph with the perspective of a Euler path or circuit and explain the following:
● In your own words, explain what is required for a path or circuit to be an Euler path or
circuit.
● Does an Euler path exist for their graph? Explain specifically using the label and degree
of each vertex.
● Does an Euler circuit exist for their graph? Explain specifically using the label and
degree of each vertex.
4) Consider your graph with the perspective of a Hamiltonian path or circuit and explain the
following:
● In your own words, explain what is required for a path or circuit to be an Euler path or
circuit.
● Identify one sequence of vertices that makes either a Hamiltonian path or a Hamiltonian
circuit.
5) Based on your real-world situation, think about whether it would be most practical to seek an
Euler vs. Hamiltonian path or circuit. Which one do you think would be more useful in your
situation and why?

Take 1+2+4+8 and continue on and on adding the next power of 2 up to infinity. This might seem crazy, but there’s a sense in which this infinite sum equals 1. If you’re like me, this feels strange or obviously false when you first see it but I promise you, by the end of this video you and I will make it make sense.
To do this, we need to back up, and you and I will walk through what it might feel like to discover convergent infinite sums, the ones that at least seem to make sense to define what they really mean, then to discover this crazy equation and stumble upon new forms of math where it makes sense.
Imagine that you are an early mathematician in the process of discovering that 1/2 + 1/4 + 1/8 + 1/16, on and on up to infinity, whatever that means, equals 1, and imagine you needed to define what it means to add infinitely many things for your friends to take you seriously. What would that feel like?
Frankly, I have no idea, and I imagine that more than anything it feels like being wrong or stuck most of the time but I’ll give my best guess at one way the successful parts of it might go.
One day you are pondering the nature of distances between objects, and how no matter how close two things are, it seems that they can always be brought a little bit closer together without touching. Fond of math as you are, you want to capture this paradoxical feeling with numbers so you imagine placing the two objects on the number line, the first at 0, the second at 1 then you march the first object towards the second such that with each step the distance between them is cut in half. You keep track of the numbers this object touches during its march writing down 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, and so on. That is, each number is naturally written as a slightly longer sum with one more power of 2 in it. As such, you are tempted to say that if these numbers approach anything, we should be able to write that thing as a sum that contains the reciprocals of every power of 2.
On the other hand, we can see geometrically that these numbers approach 1, so what you want to say is that 1 and some kind of infinite sum are the same thing. If your education was too formal, you’d write off this statement as ridiculous. Clearly, you can’t add infinitely many things. No human, computer, or physical thing ever could perform such a task. If, however, you approach math with a healthy irreverence, you’ll stand brave in the face of ridiculousness and try to make sense out of this nonsense that you wrote down, since it kind of feels like nature gave it to you.
So how exactly do you, dear mathematician, go about defining infinite sums? Well-practiced in math that you are, you know that finding the right definitions is less about generating new thoughts than it is about dissecting old thoughts, so you go back to how you came across this fuzzy discovery. At no point did you actually perform infinitely many operations. You had a list of numbers, a list that could keep going forever if you had the time, and each number came from a perfectly reasonable finite sum. You noticed that the numbers in this list approach 1, but what do you mean by “approach”? It’s not just that the distance between each number and 1 gets smaller, because for that matter the distance between each number and 2 also gets smaller.
After thinking about it, you realize what makes 1 special is that your numbers can get arbitrarily close to 1. Which is to say, no matter how small your desired distance, 1/100th, 1/1,000,000th, or one over the largest number you can write down, if you go down the list long enough, the numbers will eventually fall within that tiny tiny distance of 1. Retrospectively this might seem like the clear way to solidify what you mean by “approach”, but as a first time endeavor it’s actually incredibly clever.
Now you pull out your pen and scribble down the definition of what it means for an infinite sum to equal some number, say x. It means that when you generate a list of numbers by cutting off your sum at finite points, the numbers in this list approach x, in the sense that no matter how small a distance you choose, at some point down the list all the numbers start falling within that distance of x. In doing this, you just invented some math. But it never felt like you were pulling things out
When considering the formula for powers of two approaching zero, it is helpful to make the formula more general and applicable to all numbers. In order to do this, we must examine the definition of distance between two rational numbers. The traditional way of organizing numbers on a line may not be the only reasonable approach. A distance function should have certain properties, such as shift invariance, meaning that the distance between two numbers should not change if you shift them both by the same amount.
One way to imagine a notion of distance where powers of two approach zero is to organize numbers into rooms, subrooms, subsubrooms, and so on. For example, 0 can be seen as being in the same room as all the powers of 2 greater than 1, in the same subroom as all powers of 2 greater than 2, and so on. This hierarchy of rooms can be used to determine where numbers fall in relation to each other, including negative integers.
Turning this general idea of closeness based on rooms and subrooms into an actual distance function is a bit more complex. The size of the smallest room shared by two objects should be the determining factor in the distance between them. Using this concept, we can assign distances based on the sizes of the rooms and subrooms shared by the numbers.
This notion of distance, known as the 2adic metric, falls into a family of distance functions called the padic metrics. Each prime number (p) gives rise to a different padic metric, resulting in a new type of number that is neither real nor complex. The 2adic metric, for example, allows for the understanding of why the sum of all powers of 2 equals 1.
This process of discovering new concepts to make sense of ill-defined or nonsensical ideas is a recurring pattern in the discovery of mathematics. It is through these discoveries that rigorous terms are constructed, leading to useful math and expanding our understanding of traditional notions.