Category: Discrete Math

(Please provide detailed solutions ) . Suppose R is a reflexive and symmetric re

(Please provide detailed solutions ) . Suppose R is a reflexive and symmetric re

(Please provide detailed solutions ) . Suppose R is a reflexive and symmetric relation on a finite set A. Define a relation S on A by declaring ISy if and only if for some n € N there are elements 21, x2,…, In E A satisfying
tRa1, 71Rx2, x2Rx3,., In-Ron and In Ry.
Show that S is an equivalence relation and R C S. Prove that S is the unique smallest equivalence relation on A that contains R.
2. Consider the partition
Р = 1{0}, 1-1,1), 1-2,2), 1-3,3},…}
of Z. Describe the equivalence relation whose equivalence classes are P.
Consider the function f : Z x Z → Z defined by f(n, m) = 3m – 4n. Is this function injective?
Surjective? Bijective? Explain.
Let f: A → B be a function, and suppose Y B, then prove or disprove the following relation:
1-‘ (f(5-“(Y))) = f-1(Y)
Prove that the set of complex numbers, C, is uncountable.
Let F be the set of all possible functions R → {0,1}. Show that IR| < |F|. (Hint: Think about the relationship between F and subsets of R.)

Directions For this assignment, you will need to prove (or disprove) the followi

Directions
For this assignment, you will need to prove (or disprove) the followi

Directions
For this assignment, you will need to prove (or disprove) the following statements. You will also need to record yourself presenting the proof of each. You can do this as three videos or just one. You can record these on Zoom (or Google meet) and provide a link similar to how I do for our videos for this class (recording to a cloud). Pictures of the proofs and link(s) to your videos is what you will submit!
Your video should include the following:
A brief introduction where you show your face and introduce yourself.
A statement of the problem you are solving.
A step by step explanation of what you did to prove this statement.
A clear ending to a proof( Since the Question is asking for a video, I will need a detailed script to read from for my own purposes explaining each question on the homework)
Questions
1. Given f: A → B and subsets Y, Z C B, prove the relation below:
f-‘(Ynz) =f-‘(Y)nf-‘(z)
2. Consider the function f : R? → R? defined by the formula f(x, y) = (zy, x3). Find a formula for fo f (Nothing to prove here, just show the calculation and the steps).
3. Define a relation R on Z as xRy if and only if z? + y? is even. Prove R is an equivalence relation.
Describe its equivalence classes.

Mastermind is a game in which you have to guess a hidden code made of 4 “color p

Mastermind is a game in which you have to guess a hidden code made of 4 “color p

Mastermind is a game in which you have to guess a hidden code made of 4 “color pegs” (a tuple), from a set of 7 colors {R,G,B,Y,W,K,O}. The code maker responds to your guess with smaller “key pegs,” which are black or white (a multiset) and indicate exact or color matches with the code. (More precise definition below.)
Your assignment:
Read this guide and return to it, because I’ll update with pointers to answers with comment.
#1 Is a problem with a danger of overcounting codes with repeated colors, so I give three ways to count.
Each has an explanation from chatGPT that is often wrong, as it learns when I tell it to “Check that by brute force.” That will tell you the target number.
Make your answer self-contained by briefly restating the problem. E.g., “A guess of RRGW got a response of one black, one white key pegs. Let’s count how many codes are consistent with that response.”
question #1. You are playing Mastermind by making a guess that is a 4-tuple drawn from a set of 7 color pegs, {R,B,G,Y,W,K,O} (order matters, repeats allowed). The codemaker responds with a multiset 0 to 4 black or white key pegs (order doesn’t matter; repeats allowed). Each b indicates some guess peg matches the code in position and color; w indicates some guess peg has a right color in the wrong position. (A more precise statement of the rules is in CW#30.)
You want to know how many codes are consistent with the response to your guess, because that is the first step to narrowing this down to one code.
Here is your guess and the response:
RRRYww
https://chat.openai.com/share/ea81a6de-48cd-442f-8…
question 2: You are playing Mastermind by making a guess that is a 4-tuple drawn from a set of 7 color pegs, {R,B,G,Y,W,K,O} (order matters, repeats allowed). The codemaker responds with a multiset 0 to 4 black or white key pegs (order doesn’t matter; repeats allowed). Each b indicates some guess peg matches the code in position and color; w indicates some guess peg has a right color in the wrong position. (A more precise statement of the rules is in CW#30.)
You want to know how many codes are consistent with the response to your guess, because that is the first step to narrowing this down to one code.
Here is your guess and the response:
RGBKw: https://chat.openai.com/share/aed9a55d-12d8-402a-9…
Please restate the guess and response to make your answer self-contained. You are encouraged to find different solutions from existing answers, or present it in a different way (with diagrams, to a bright middle schooler, as briefly as possible, …) or diagnose what the linked ChatGPT transcript gets right or wrong. (It’s “check by brute force” does report the right number, though its explanation is often wrong.)

Mastermind is a game in which you have to guess a hidden code made of 4 “color p

Mastermind is a game in which you have to guess a hidden code made of 4 “color p

Mastermind is a game in which you have to guess a hidden code made of 4 “color pegs” (a tuple), from a set of 7 colors {R,G,B,Y,W,K,O}. The code maker responds to your guess with smaller “key pegs,” which are black or white (a multiset) and indicate exact or color matches with the code. (More precise definition below.)
Your assignment:
Read this guide and return to it, because I’ll update with pointers to answers with comment.
#1 Is a problem with a danger of overcounting codes with repeated colors, so I give three ways to count.
Each has an explanation from chatGPT that is often wrong, as it learns when I tell it to “Check that by brute force.” That will tell you the target number.
Make your answer self-contained by briefly restating the problem. E.g., “A guess of RRGW got a response of one black, one white key pegs. Let’s count how many codes are consistent with that response.”
question #1.
You are playing Mastermind by making a guess that is a 4-tuple drawn from a set of 7 color pegs, {R,B,G,Y,W,K,O} (order matters, repeats allowed). The codemaker responds with a multiset 0 to 4 black or white key pegs (order doesn’t matter; repeats allowed). Each b indicates some guess peg matches the code in position and color; w indicates some guess peg has a right color in the wrong position. (A more precise statement of the rules is in CW#30.)
You want to know how many codes are consistent with the response to your guess, because that is the first step to narrowing this down to one code.
Here is your guess and the response:
RRRYww
https://chat.openai.com/share/ea81a6de-48cd-442f-8…
question 2:
You are playing Mastermind by making a guess that is a 4-tuple drawn from a set of 7 color pegs, {R,B,G,Y,W,K,O} (order matters, repeats allowed). The codemaker responds with a multiset 0 to 4 black or white key pegs (order doesn’t matter; repeats allowed). Each b indicates some guess peg matches the code in position and color; w indicates some guess peg has a right color in the wrong position. (A more precise statement of the rules is in CW#30.)
You want to know how many codes are consistent with the response to your guess, because that is the first step to narrowing this down to one code.
Here is your guess and the response:
RGBKw: https://chat.openai.com/share/aed9a55d-12d8-402a-9…
Please restate the guess and response to make your answer self-contained. You are encouraged to find different solutions from existing answers, or present it in a different way (with diagrams, to a bright middle schooler, as briefly as possible, …) or diagnose what the linked ChatGPT transcript gets right or wrong. (It’s “check by brute force” does report the right number, though its explanation is often wrong.)

Set Theory as a Framework for Relational Databases A set can be a collection of

Set Theory as a Framework for Relational Databases
A set can be a collection of

Set Theory as a Framework for Relational Databases
A set can be a collection of any type of object, ranging from people to places to things. Basic set theory includes the study of subsets, proper subsets, finite and infinite sets, and the logical operations on them. Set theory plays a foundational role in mathematical processes and ideas and also has connections to computer engineering, programming, and databases.
The relational database model, originally invented by computer scientist Edgar F. Codd in 1969, is based on ideas from set theory. A simple database is a collection of records stored in tables. A relational database also includes relationships stored across multiple tables. One can run queries on the relational database to request specific information with set theory operators, such as union and intersection.
Post 1: Initial Response
Imagine you are responsible for your organization’s analytic tasks, and you are currently brainstorming how to query a relational database of marketing information for the organization. You want to test your understanding of how you might relate the database tables with the use of set theory, and particularly subsets. To carry out your test, complete each of the following:
To define two sets, set A and set B, first conduct an online browsing trial, in which you spend 10–20 minutes looking at different websites, such as for national news, social media, sports, hobbies, recipes, etc. Let set A represent exactly three distinct company names from any online advertisements you saw during your browsing trial. Let set B represent at least three distinct company names for any online retailers you have purchased from in the past year.
To prepare to use your algorithm, answer the following questions:How many elements are in set A? This is what you will set as m = ___.
How many elements are in set B? This is what you will set as n = ___.
What are your first and last elements of A? Show these as a[1] = ____ and a[m] = ___.*
What are your first and last elements of B? Show these as b[1] = ____ and b[n] = ___.*
Using your sets A and B along with what you just outlined to prepare, determine an algorithm that you can use to see whether A ⊆ B. You can make your own or find one somewhere else.
State the algorithm that you would use to compare these sets. If you are using an algorithm that you did not write, cite or describe where you found it.
Based on your algorithm, did you find that A ⊆ B or that A ⊈ B? Explain. If A ⊈ B, how are they related (e.g., disjoint, intersecting)?
View Unit 6 Discussion Post 1 example.
Post 2: Reply to a Classmate
Now you want to try out the algorithm on another person’s data to further test your understanding and bolster your confidence about assessing relations computationally as you approach this relational database project.
Review a classmate’s post and consider their set B. Address the following items completely.
Using your set A and their set B, use the algorithm they described to determine whether A ⊆ B or A ⊈ B? Explain how you know this. If A ⊈ B, how are they related (e.g., disjoint, intersecting)?
How might the understanding you have gained from your Post 1 and Post 2 tests be useful if you were responsible for querying a relational database?
View Unit 6 Discussion Post 2 example.
Post 3: Reply to Another Classmate
After conducting this computational practice, you have begun to develop some technical insight into how you might investigate and seek information on the marketing habits of clients by querying a relational database. However, you know your fellow staff members are not interested in this technical insight. So, for your general meeting, you plan to present a visual synopsis of some ideas considered in the planning stages of this project.
Review another classmate’s post and consider their sets A and B. Address the following items completely.
Create a Venn diagram that models all of the elements in your classmate’s sets A and B. Carefully place elements appropriately in the intersecting versus non-intersecting areas representing sets A and B, respectively. You may use the software of your choice for the Venn diagram (e.g., creatly.com, cosketch.com, Microsoft® Word®, or PowerPoint®). Copy and paste the image or screenshot of your Venn diagram into your post. (You may also use an attached file if needed.)
Draft some talking points in anticipation of addressing the following questions during your presentation:How do these two sets relate in the example illustrated by the Venn diagram?
How have the concepts of sets and set operations been utilized in your analytic tasks?
How might table relationships be modeled from the ideas of set theory?
View Unit 6 Discussion Post 3 example.