Discrete math

Discrete math

Exercise A

Write down, in your own words, a detailed specification for a library loans and returns record system

You are required to:

• Fully specify your systems using propositions and predicates
• Test the validity of a simple propositional clause, relevant to the library by constructing a truth table

Exercise B

For each of these binary relations, determine whether they are reflexive/irreflexive, symmetric/antisymmetric or transitive. Give reasons for your answers and state whether or not they form order relations or equivalence relations.

1. On the set {alf, bert, tom}, the relation

{(alf,alf), (alf, bert), (bert, bert), (tom,tom) }.

2. Take E to be the set of all arithmetic expressions involving variables such as x, y, z, For example, “x +3y” is an expression belonging to E. The binary relation on E contains all pairs of expressions, (exp1, exp2), which give same result for all values of their variables. For example, the pair of expressions (“x+3y”, “y+x+2y”) is in the relation because they have the same values for all values of x and y. But (“x+3y”, “x”) does not belong to the relation as the expressions have different values when x=1 and y=1, say.

For the following question, state the relation and explain your reasoning.

3. Let W be the set of days in the week {Sun, Mon, Tue, Wed, Thu, Fri, Sat}. What relation, as a subset of WxW, corresponds to the partition that has three equivalence classes and treats Sat and Sun as equivalent, Tue and Thu as equivalent, and Mon, Wed, Fri as equivalent?

Exercise C

6. The sine and cosine functions are defined on real numbers and represent waves that have a cycle of length about 6.3 (2π radians, equivalent to 360°).

Thus, sin(x) completes a cycle in about 6.3 units.

Use Excel to investigate the use of sine and cosine functions. Construct a column containing values of x from 0 to 20 in steps of 0.1. Produce a column containing sin(x) for each value of x, and plot its curve.

Look at sin(2x), sin(3x) and sin(5x). Plot their curves and comment on the length of their cycles.

Do the same for cos(x), cos(2x), cos(3x) and cos(5x).

Investigate the effect of adding multiples of these functions, such as

cos(x) + 2sin(3x) + 4 cos(5x)

and two further examples of your own, plotting the curves for each.

Write a short report, including copies of your plotted curves, to explain what you have discovered about these functions.

1. In Windows Media Player you can set a visualization mode to show a Scope (in the Bar and Waves group) when you play audio. Using the Pause button during the play will freeze the display for you to examine more closely.

Using two or three pieces of music of different types, examine the waves produced by the music and capture four or five examples

Write a brief report to discuss how well you think functions such as those you used in part i are able to model these audio waves produced by your chosen types of music.

Exercise C

6. The sine and cosine functions are defined on real numbers and represent waves that have a cycle of length about 6.3 (2π radians, equivalent to 360°).

Thus, sin(x) completes a cycle in about 6.3 units.

Use Excel to investigate the use of sine and cosine functions. Construct a column containing values of x from 0 to 20 in steps of 0.1. Produce a column containing sin(x) for each value of x, and plot its curve.

Look at sin(2x), sin(3x) and sin(5x). Plot their curves and comment on the length of their cycles.

Do the same for cos(x), cos(2x), cos(3x) and cos(5x).

Investigate the effect of adding multiples of these functions, such as

cos(x) + 2sin(3x) + 4 cos(5x)

and two further examples of your own, plotting the curves for each.

Write a short report, including copies of your plotted curves, to explain what you have discovered about these functions.

1. In Windows Media Player you can set a visualization mode to show a Scope (in the Bar and Waves group) when you play audio. Using the Pause button during the play will freeze the display for you to examine more closely.

Using two or three pieces of music of different types, examine the waves produced by the music and capture four or five examples.

Write a brief report to discuss how well you think functions such as those you used in part i are able to model these audio waves produced by your chosen types of music.

Exercise D

Explain how a compiler might use a tree structure to represent an arithmetic expression and generate the low-level instructions equivalent to it.

Show the trees it would use for the expressions

(x + y) / ((x – y) + z)

z + (y * x) – (x + z)

and evaluate the trees when x=6, y=3 and z=2.

Suppose there are a number of modules to schedule for lectures using as few time slots as possible. If two modules are shared by a group of students, then they cannot be scheduled in the same time slot. Show that this problem can be solved by colouring a graph (using as few colours as possible) so that no two vertices connected by an edge have the same colour. The following table shows the set of modules taken by groups of students.

Show that the above modules can be scheduled in three time slots by colouring the corresponding graph with three colours.

Exercise E

The module has examined modelling in computing to show how the concepts from set theory and logic, functions and relations, trees and graphs underlie the modelling of computer systems and programs.

Write a report of between 3 and 6 A4 pages to discuss the topics of your portfolio and what they show about the importance of the concepts in modelling aspects of systems in computing:

You should comment on what you have learned from this portfolio of exercises

Suggest some further aspects of computing systems that may be modelled with the tools and techniques you have learnt.