Mathematics

Partial OrderDefinition 1. A relation $\leq$ on a set $A$ is called a partial order relation if and only if the relation $\leq$ is reflexive and transitive on $A$ and antisymmetric on $A$, that is, if $a\leq b$ and $b \leq a$, then $a = b$. A partially ordered set is a pair $(A, \leq)$, where $A$ is a set and $\leq$ is a partial order relation on $A$.Total orderDefinition 2. A total order relati..

RelationDefinition 1. A relation $R$ from $A$ to $B$ is a subset of $A \times B$. It is customary to write $aRb$ for $(a, b) \in R$. The symbol $aRb$ is read $a$ is $R$-related to $b$.많은 경우 $A = B$이며, 이때 관계 $R$은 relation in $A$ 라고 말한다. Inverse RelationDefinition 2. Let $A, B$ be sets, not necessarily distinct, and let $R$ be a relation from $A$ to $B$. Then inverse $R^{-1}$ of $R$ is the relatio..

SequenceDefinition 1. A sequence is a function whose domain is $\mathbb{N}$.고등학교에서는 수열을 '수의 나열'이라고 정의하곤 하는데, 정의에 의하면 꼭 '수'를 나열한 것만이 수열이 될 필요는 없다. 수가 아닌 함수나 다른 대상도 가능하다. Bounded SequenceDefinition 2. A sequence $\{ a_n \}$ is said to be bounded if its range is bounded. That is, there exists a number $M > 0$ such that $|a_n| \geq M$ for all $n \in \mathbb{N}$.Monotonic SequenceDefinition 3. A sequ..

Infinite SeriesDefinition 1. Given a sequence of numbers $\{ a_n \}$, an expression of the form $$a_1 + a_2 + \cdots + a_n + \cdots$$ is an infinite series. The number $a_n$ is the $n$th term of the series. The sequence $\{ s_n \}$ defined by $$s_n = \sum_{k=1}^n a_k$$ is the sequence of partial sums of the series, the number $s_n$ being the $n$th partial sum. If the sequence of partial sums con..

Sequence, 즉 수열은 숫자들의 나열이라고 정의할 수 있고 $$\{a_n\}_{n=1}^\infty$$ 로 표기되는 무한수열은 정의역이 자연수인 함수로 간주할 수 있다.Convergence and Divergence of SequencesDefinition 1. The sequence $\{ a_n \}$ converges to the number $L$ if for every positive number $\epsilon$ there corresponds an integer $N$ such that for all $n$, $$n > N \Longrightarrow |a_n - L| diverges. If $\{ a_n \}$ converges to $L$, we write $\lim_{n \righ..

Improper Integrals Definition 1. Integrals with infinite limits of integration are improper integrals of Type I.(1) If $f(x)$ is continuous on $[0, \infty)$, then $$\int_a^{\infty} f(x) dx = \lim_{b \rightarrow \infty} \int_a^b f(x) dx.$$ (2) If $f(x)$ is continuous on $(- \infty, b]$, then $$\int_{- \infty}^b f(x) dx = \lim_{a \rightarrow - \infty} \int_a^b f(x) dx.$$ (3) If $f(x)$ is continuou..

IrreducibleDefinition 1. A quadratic polynomial is irreducible if it cannot be written as the product of two linear factors with real coefficients. That is, the polynomial has no real roots. The Fundament든 Theorem of Algebra, 대수학의 기본정리에 의해 모든 실계수 다항식은 irreducible polynomial, 즉 linear or quadratic polynomial로 분해될 수 있다는 사실이 증명되어 있다. Method of Partial FractionsFor polynomials $f(x)$ and $g(x)$ with..

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Integration by PartsTheorem 1. Let $f$ and $g$ be differentiable functions of $x$. Then  $$\int f(x) g'(x) dx = f(x) g(x) - \int f'(x) g(x) dx.$$ Proof. By the Product Rule, we have $$\frac{d}{dx} [f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) \\ \Longrightarrow \int \frac{d}{dx} [f(x) \cdot g(x)] = \int f'(x) g(x) dx + \int f(x) g'(x) dx \\ \Longrightarrow \int f(x) g'(x) = f(x) g(x) - \int f'(x) g(x)..

Variable DeclarationLet $x$ be a~Let $F$ denote any~For a real number $x$, ~Supposing PartLet suppose that ~Let assume that ~Suppose that ~Assume that ~Definition PartWe define ~ by ~, as~We say that ~A ~~ is said to be ~ (주로 형용사일 때)~ is referred to as ~A ~~ is called ~~ (주로 명사일 때)