Mathematics Articles

Mathematics Articles - Scientist Adding Microscopy To Brain Study App

Mathematics

Mathematics Articles Surprisingly, mathematics has no widely accepted definition. It has been defined in various ways for centuries, such as the science of quantity, logic, or intuition. A few main schools of thought exist today about how to formally define it, but let’s leave that for more in-depth mathematics articles. In a nutshell, you can …

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Confluence Relations

Confluent Relations (using Reduction Relations)

We discuss confluent relations; in particular, we prove Newman’s Lemma: that local confluence, confluence, the Church-Rosser property, and the unique normal forms property are all equivalent for a well-founded relation. We also give a generalization of Newman’s lemma based on the Buchberger-Winkler’s Property. Reduction Relations Let $\longrightarrow$ be a relation on $X.$ If there exists  …

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Well-Founded Relations

Well-Founded Relations (and Well-Founded Induction)

Well-Founded induction is a generalization of mathematical induction. Well-Founded Induction Definition. Let $\longrightarrow$ be a relation on $X.$ 1) If $A\subseteq X$ and $a\in A,$ then $a$ is called a $\longrightarrow$-minimal element of $A$ if  there does not exist $b\in A$ such that $a\longrightarrow b.$ 2) If each nonempty subset of $X$ has a $\longrightarrow$-minimal …

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Equivalence Relations

Equivalence Relations (Properties and Closures)

Equivalence Relations We discuss the reflexive, symmetric, and transitive properties and their closures. The relationship between a partition of a set and an equivalence relation on a set is detailed. We then give the two most important examples of equivalence relations. Reflexive, Symmetric, and Transitive Relations Definition. Let $X$ be a set. A relation $R$ …

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One-to-One Functions and Onto Functions

One-to-One Functions and Onto Functions

One-to-one and Onto Functions Injective Functions Definition. Let $X$ and $Y$ be sets. A function $f:X\to Y$ is called injective if  $$ \forall x_1,x_2\in X, f(x_1)=f(x_2)\implies x_1=x_2. $$ Theorem. Let $f:X\to Y$ be a function.  If $f$ is injective and $A\subseteq X$, then $f|_A$ is injective. Proof. Let $x_1, x_2\in A$.  Then  $$ f|_A(x_1)=f|_A(x_2)  \Longrightarrow …

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Set theory

Set Theory (Basic Theorems with Many Examples)

We discuss the basics of elementary set theory including set operations such as unions, intersections, complements, and Cartesian products. We also demonstrate how to work with families of sets. For a brief discussion of the reviews of (elementary) Halmos’ Naive Set Theory read this. A solid understanding of propositional and predicate logic is strongly recommended. To …

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Probability Density Functions

Probability Density Functions (Applications of Integrals)

Applications of Integrals We will consider the following applications: average value of a function over a region, mass of a lamina, electric charge, moments and center of mass, moments of inertia, and probability density functions. Average Value Recall the average value of an integrable function of one variable on a closed interval is the integral …

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Divergence and Curl

Divergence and Curl of a Vector Field

The Divergence and Curl Definition. Let $\mathbf{V}$ be a given vector field. The divergence of $\mathbf{V}$ is defined by div $\mathbf{V}=\nabla \cdot \mathbf{V}$ and the curl of $\mathbf{V}$ is defined by curl $\mathbf{V}=\nabla \times \mathbf{V}$ where \begin{equation} \nabla =\frac{\partial }{\partial x}\mathbf{i}+\frac{\partial }{\partial y}\mathbf{j}+\frac{\partial }{\partial z}k\end{equation} is the del operator. For a discussion on the physical …

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Determinant of a Matrix

Determinant of a Matrix (Theory and Examples)

Determinants and Trace The determinant function can be defined by essentially two different methods. The advantage of the first definition, one which uses permutations, is that it provides an actual formula for $\det(A)$, a fact of theoretical importance. The disadvantage is that, quite frankly, computing a determinant by this method can be cumbersome. A pattern …

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