Challenge Task 1 - Variant PDF

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MAT1252D

Challenge Task 1

Challenge Task 1 Updated: 16 Apr 2021

Objectives The objectives of this task are: •

To use the skills you have developed in MAT1252D thus far to find solutions to the provided problems. You should make use of any appropriate mathematical techniques you have encountered in MAT1252D to develop your solutions.

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To write and include in your submission a single-page postmortem for this task using the provided template.

Task Submission This task is due 24 hours after it is assigned. All submissions should be conducted electronically through Moodle. Your attempt should be submitted as a single zip archive. Please DO NOT submit rar or 7zip archives as doing so may result in a mark of zero for the task. Your submission should contain, at a minimum: •

Your model problems and solutions to your assigned task, in one of the following formats: ◦ Scanned handwritten solutions combined into a PDF ◦ Photographed handwritten solutions combined into a PDF ◦ A series of photographs of your work

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Your postmortem document in PDF format.

You can find your individualised task on the next page. Be sure to include your name and student number in the top right corner of each page. DO NOT PRODUCE WORD PROCESSED SOLUTIONS.

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MAT1252D

Challenge Task 1

Your Task – Part A 1. Complete the following tasks related to the Week 4 content: a. Construct a Karnaugh map for the following Sum of Products: x’y + x’z +xyz’

b. Construct a minimal Sum of Products based on the following Karnaugh map: 1 1

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1

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c. Draw the circuit corresponding the the MinSoP you derived in part b above.

2. Complete the following tasks related to Week 5 content: a. Consider two sets A = {a, b, c ,d} and B = {x, y, z}. Let R be the relation defined by: R = {(a, x), (a, y), (b, x), (c, z), (d, x), (d, z)} Draw the graph of R, and the arrow diagram for R.

b. Express the relation “is the cube of” defined on the set {-8, -4, -2, -1, 1, 2, 4, 8} as a set of ordered pairs.

c. Consider sets H = {a, b, c}, K = {d, e}, and L = {f, g, h, k}, and relations U between H and K, and V between K and L, defined by: U = {(a, d), (a, e), (b, e), (c, e)} V = {(d, h), (d, k), (e, g), (e, h), (e, k)} Draw the arrow diagram representing the composition of relations V and U, and hence give the set of ordered pairs in that composition.

3. Complete the following tasks related to Week 6 content: a. Consider the function f: {1, 3, 5, 7, 9} →R, f(x) = x(x – 5) Find the value of f(3), and show that it is part of the range of f.

b. Suppose that f(x) = 3x + 1, and g(x) = 3x – 2 Find an expression for f(g(x)), and thus calculate f(g(4)) TASK CONTINUES ON NEXT PAGE Page 2 of 3

MAT1252D

Challenge Task 1

Your Task – Part B Your task will be to design mathematical problems and model solutions suitable for use in the unit MAT1252D according to a provided specification, and covering a variety of topics. As such, you are required to attempt the following: 1. Create a Karnaugh map designed to be expressed as a Complete Sum of Products and also as a truth table. 2. Specify sets and relations such that the model solution can be represented as both an arrow diagram and a graph. 3. Given a domain, produce a codomain and series of relations that are functions designed to illustrate the concepts of both one-to-one and onto functions. Produce a problem or series of problems and a model solution or solutions for each question (starting below), and include any drawings you need to assist you. Be sure to list any additional information needed by somebody who might attempt your problems as part of your answer. 1) Create a maths problem and model solution corresponding to the following question: “Express the following Karnaugh map as both a Complete Sum of Products and a Truth Table” Use 4 boolean values, and label them e, f, g, and h. Your Karnaugh map should have five “1”s in it, one of which should correspond to e’f’g’h.

2) Create a maths problem and model solution corresponding to the following question: “Draw an arrow diagram and also produce a graph to represent the following relation” Your problem should outline 2 sets, and label them E and K. Set K should have 4 elements. Set E should have elements labelled e, f, g, h. Create a relation, labelled R, with at least 7 pairs.

3) Create a group of 4 maths problems and model solutions corresponding to the following question: “Consider the following 4 functions. For each, state whether it is one-to-one, onto, both, or neither” Your problem should outline 4 sets of ordered pairs based on two sets that you should label “F”, and “J”. Set “F”, defined as {b, c, d, e, f, g} will be the domain. “J” will be the codomain, and may vary by function. Label each of the sets of ordered pairs s, t, u and v respectively. Set “v” should include the ordered pair (b, m). Demonstrate at least one function that is only one-to-one, one that is only onto, one that is both one-to-one and onto, and one that is neither. Provide diagrams as part of your model solutions. END OF TASK

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