This website houses notes from my studies at the University of Liverpool. If you see any errors or issues, please do open an issue on this site's GitHub.
$n$-vectors A set of numbers, $H$. $H$ is one of: $\Bbb{N,W,Z,Q;R}$. A natural number, $n\in\Bbb{N}$. An $n$-vector over $H$, has $n$ components: \(\underline x = \langle x_1,x_2,\ldots,x_n\rangle\) Each $x_k\in H$. This is not a set: we cam have $x_s=x_t$ and $s\neq t$. The order of components matters: $\langle 1,2,1\rangle\neq\langle1,1,2\rangle$. Notation...
Irrationals Revisited Irrational numbers such as $\sqrt{2}$ can be seen as solutions to particular identities. Polynomials & Their Roots In general, a polynomial of degree $k$ in $x$ is defined through $k+1$ coefficients: \[(c_0,c_1,\ldots,c_k)\] This can be written in the following functional form: \[p(x)=\sum^k_{n=0}c_nx^n\] The coefficient, $c_t$, multiplies $x^t$ when...
What are Numbers? Numbers are the objects that we use to count things. Depending on what we are counting we want to use different sets, or types, of numbers. Sets of Numbers $\Bbb N$:1,2,3,4… (Natural numbers) Zero is not a part of this set. $\Bbb W$:0,1,2,3,4… (Whole numbers) $\Bbb Z$:...
This lecture covers the answers to last year’s class test which is available here. 2019 Class Test Question 6 Tail recursion is when once you have recursed there is nothing left to process. Having an accumulator is a sign that this is the case.
This lecture was given in the form of a PDF. View it here if you want to see it in full. There are several schemes for protecting intellectual property. Patents Form of legal protection given to individuals who create an invention or process. Software can be patented in some countries,...
This lecture was given in the form of a PDF. View it here if you want to see it in full. Linked to this lecture, there is a paper on Ethical Interest in Free and Open Source Software. GNU and FSF FSF was formed in 1985 to support of the...
This lecture was given in the form of a PDF. View it here if you want to see it in full. Property Tangible Property Tangible property originally referred to land but now refers to any physical object that a person can own. Relational Property As property becomes less physical we...
Binomial Coefficients The quantity $C(n,k)$, which gives the number of $k$-combinations of a set of size $n$, is called a binomial coefficient. It is also written as: \[{n\choose k}=\frac{n!}{(n-k)!k!}\] This notation is preferred. Binomial Theorem For every natural number $n$: \[(a+b)^n=\sum^n_{k=0}{n\choose k}a^kb^{n-k}\] This is the sum of a generic binomial...
Combination Examples The number of size 1 subsets of $\{1,2,3,4,5\}$ is 5. This is also the number of size 4 subsets of the set. Any opposite subset sizes sample complements of each other and are therefore the same size. Twelve people, including Mary and Peter, are candidates to serve on...
We can view cyberethics as a branch of three possible perspectives. Professional Ethics According to this view cyberethics is the responsibility of the professionals. This also means that the issues aren’t strictly about the technology itself. This perspective is not sufficient as the issues don’t just affect professionals or even...