Matrices and System of Linear Equations Let $a_x$ be scalars. We can view a sequence or array $S=\langle a_1,\ldots,a_n\rangle$ as a function $f_S:\{1,\ldots,n\}\mapsto F$ where $f_S(i)=a_i$. Similarly a matrix, $M$ is a 2$d$ array which can be seen as a $f_M:\{1,\ldots,m\}\times\{1,\ldots,n\}\mapsto F$ where $f_M(i,j)=a_{i,j}$. We define product of matrix $A$ and $B$ as follows $$C=AB\coloneqq C_{ij}=\sum^n_{r=1}A_{ir}B_{rj}$$ This definition does not look interesting by itself. We need an application to appreciate its definition and that application comes along with the matrices in representing linear transformation of vector spaces. ...
Linear Algebra - Part 1
Linear algebra is the study of vector spaces and transformation between these vector spaces. A naive view of vector would be of an $n$-tuple of scalar (which are some objects like real numbers), If $F$ is our set of scalars then an $n$-dimensional vector is an element of $F^n. It is popular to write vectors as $\vec{v}$, where is supposed to denote an arrow pointing to some point in an Euclidean like space from the origin. Though such view of vector space being some algebra in Euclidean space akin to coordinate geometry is prevalent, it is but one particular case of vector spaces. We will see over time that it is indeed possible to have a one to one correspondence between any vector space to $F^n$, and yet we should keep in mind that vectors are abstract objects that could be anything satisfy its axioms. ...
Formal Logic - Part 3
Deductive Calculus Deductions are formal proofs Given a set of formula $\Lambda$ called logical axioms, the theorems of a set of formula $\Gamma$ are formulas obtained from $\Lambda\cup\Gamma$ by using the rules of inference. The choice of the rules of inference and $\Lambda$ are arbitrary. An example for the choice of rules of inference Modus ponens: $\alpha,\alpha\rightarrow\beta\vdash\beta$ A deduction of $\varphi$ from $\Gamma$ is a finite sequence $\langle a_1,\ldots,a_n\rangle\ni a_n=\varphi\land\forall k\le n:$ $a_k\in\Gamma\cup\Lambda$ or $\exists\ i<k,j<k:\alpha_j=\alpha_i\rightarrow\alpha_k$ If a deduction of $\varphi$ from $\Gamma$ exists, we say $\varphi$ is deducible from $\Gamma$ or that $\varphi$ is a theorem of $\Gamma$, and we write it as $\Gamma\vdash\varphi$ Rules of inference as an operation may not be totally defined or generate theorems freely. $\mathrm{wff}\ \varphi$ is a generalization of $\psi\ \mathrm{iff}$ $\varphi=\forall x_1\ldots\forall x_n\psi$ Prime formulas are either atomic formulas or formula of the form $\forall\ x\ \varphi$. Tautologies in FOL are obtained by replacing sentential symbols in tautology in sentential logic with prime formulas. $\mathrm{wff}$ in FOL are also $\mathrm{wff}$ in sentential logic. Substitution of $x$ by $t$ in $\alpha$, $\alpha^x_t$ is defined as: For an atomic $\alpha$, replace occurrences of $x$ with $t$ $(\alpha\rightarrow\beta)^x_t=(\alpha^x_t\rightarrow\beta^x_t)$ $(\forall\ y\ \alpha)^x_t=\begin{cases}\forall\ y\ \alpha&&\text{if }x=y\\forall\ y(\alpha^x_t)&&\text{if }x\ne y\end{cases}$ An example for the choice of $\Lambda$: Tautologies $\forall\ x\ \alpha\rightarrow\alpha^x_t$ where $t$ is substitutable for $x$ in $\alpha$ $\forall\ x\ (\alpha\rightarrow\beta)\rightarrow(\forall \ x\ \alpha\rightarrow\forall\ x\ \beta)$ $\alpha\rightarrow\forall\ x\ \alpha$ where $x$ does not freely occur in $\alpha$ $x=x$ $x=y\rightarrow(\alpha\rightarrow\alpha’)$ where $\alpha’$ is an atomic obtained from $\alpha$ by replacing some or all $x$ with $y$ $\Gamma\models\varphi\longrightarrow\Gamma\vdash\varphi$. Since modes ponens is a tautological implication, with modus ponens as the rule of inference, we have $\Gamma\models\varphi\longleftrightarrow\Gamma\vdash\varphi$
Formal Logic - Part 2
The Language of FOL We will now look at a more powerful system of formal logic called first order logic. What makes it more powerful than sentential logic is use of variables whose domain is decided by quantifiers. Since it is a formal system again, we must define the symbols and rules for the language. Symbols We have sentential connectives and parenthesis just like we had in sentential logic. Besides these, we need three other symbols which are: ...
Formal Logic - Part 1
We can call formal logic the entirety of maths. We start based on some assumed information, and we try to produce new information from it. This process of transforming information is called reasoning and the steps invloved while doing so is called inference. Logic is the study of this reasoning. Formal languages are collection of strings, string itself being a sequence of symbols. Automatas are state machines with rules on how those state changes while processing an input, and these state machines can be used to decide if a string belongs to a particular language. Formal logic uses this idea of formal languages and automata theory to study reasoning. ...