Mathematics

Partition Definition 1. For a closed interval $[a, b]$, we subdivide the interval into subintervals, not necessarily of equal widths, as choosing $n-1$ points $\{ x_1, ..., x_{n-1} \}$ between $a$ and $b$ that are in increasing order, so that $x_0 = a partition of$[a, b]$. And we denote the width of the$k$th subinterval by$\Delta x_k$which means that$\Delta x_k = x_k - x_{k-1}$. Norm of a P..

Rolle's TheoremTheorem 1. Let $f$ be a function that satisfies the following conditions:(1) $f$ is continuous on $[a, b]$.(2) $f$ is differentiable on $(a, b)$(3) $f(a) = f(b)$Then there is a number $c \in (a, b)$ such that $f'(c) = 0$.Proof. We may think of three cases. (1) If $f(x) = k$ for any constant $k$, then $c$ can be taken to be any number in $(a, b)$.(2) If $f(x) > f(a), \forall x \in ..

Fermat's TheoremTheorem 1. If $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$.Proof. Without loss of generality, suppose that $f$ has a local maximum at $c$. This means that $f(c) \geq f(c+h)$ for $h$ which is sufficiently close to $0$. If $h > 0$, we have $$\frac{f(c+h) - f(c)}{h} \leq 0 \\ \Longrightarrow \lim_{h \rightarrow 0^+} \frac{f(c+h)-f(c)}{h} = f'(c)..

Absolute Maximum and MinimumDefinition 1. Let $c$ be a number in the domain $D$ of a function $f$. Then $f(c)$ is the(1) absolute maximum value of $f$ on $D$ if $f(c) \geq f(x), \forall x \in D$.(2) absolute minimum value of $f$ on $D$ if $f(c) \leq f(x), \forall x \in D$.$f$의 maximum과 minimum은 extreme value of $f$, 즉 $f$의 극값이라고 부르기도 한다.  Local Maximum and MinimumDefinition 2. The number $f(c)$ ..

Conditional ProbabilityDefinition 1. Let $E$ and $F$ be events. We define the conditional probability that $E$ occurs given that $F$ has occurred, denoted by $P(E | F)$, by $$P(E | F) = \frac{P(EF)}{P(F)}$$ if $P(F) > 0$.사건 $F$가 먼저 일어났다는 가정 하에 $E$가 일어나는 확률을 위와 같은 방법으로 정의한다. 이때 sample space를 $F$로 한정 지을 수 있고, $F$와 동시에 $E$가 일어나야 하므로 위와 같은 정의는 합리적이다. 위 식에서 양변에 $P(F)$를 곱함으로써 $$P(EF) = P(F)P(E | F)$$ 로..

Axioms of ProbabilityLet $E \subset S$ be an event. Then we call the function $P : E \rightarrow \mathbb{R}$, following below conditions, the probability. (1) $$0 \leq P(E) \leq 1$$ (2) $$P(S) = 1$$ (3) For any sequence of mutually exclusive events $E_1, E_2, ...$, $$P(\bigcup_{i=1}^{\infty} E_i) = \sum_{i=1}^{\infty} P(E_i)$$ 통계적인 의미에서 확률은 relatvie frequency, 상대 빈도수의 관점에서 정의되었다. 즉, sample space가 $..

Union and IntersectionDefinition 1. For any events $E, F$, we define the union of $E$ and $F$ by $E \cup F$. Similarly, the intersection of $E$ and $F$ is defined as $EF = E \cap F$. 두 사건 $E, F$가 있을 때 이 사건 둘 중 하나가 혹은 둘 다 일어나는 사건을 union, 합사건이라고 하고, 둘 다 일어나는 사건을 intersection, 곱사건이라고 부른다. 숫자를 늘려서 다음과 같이 무한합, 무한곱도 정의할 수 있다. Definition 2. Let $E_1, E_2, ...$ be events. We define the union of these ev..

Sample SpaceDefinition 1. Sample space is the set of all possible outcomes of an experiment, and is usually denoted by $S$.모든 시행에서 나올 수 있는 결과값을 모아놓은 집합을 sample space, 표본 공간이라고 부른다.  EventDefinition 2. Any subset $E$ of the sample space is called an event. In other words, an event is a set consisting of possible outcomes of the experiment. If the outcome of the experiment is contained in $E$, the..

The Basic Principle of CountingThe Basic Principle of Counting. If an experiment consisting of two phases is such that there are $n$ possible outcomes of phase 1 and, for each of these $n$ outcomes, there are $m$ possible outcomes of phase 2, then there are $nm$ possible outcomes of the experiment. '곱의 법칙'으로 흔히들 배우는 내용이다. 첫 번째에서 $n$개의 케이스가 나오고, 각 케이스에 대해서 두 번째에는 $m$개의 케이스가 존재한다면 총 $nm$개의 케이스가 존재..

Theorem 1. A set $O$ is open $\iff$ $O^c$ is closed. Likewise, a set $F$ is closed $\iff$ $F^c$ is open.Proof. $(\Longrightarrow)$ $O$가 open이라고 하자. $x$를 $O^c$의 limit point라고 하면, $x$의 어떤 근방을 가지고 와도 항상 $O^c$와 겹치는 부분이 존재하므로 $x \in O^c$이다. ($\because$ 만약 $x \in O$이면 $x$의 어떤 근방이든 항상 $O$에 완전히 포함되므로 $O^c$와 겹치는 부분이 생길 수 없다.) 따라서 $O^c$는 closed이다.$(\Longleftarrow)$ $O^c$가 closed라고 하자. $a \in O$를 가져온 뒤 $a$..