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Random math: 07

Source of most content: Gunhee Cho Ordered set Def. Given a set $S$, we call $\leq$ partial order and call $(S, \leq)$ partially ordered set (POSET) if $\leq$ satisfying: (Reflexibility) $\forall a \in S, a \leq a$. (Anti-symmetry) If $a \leq b$ and $b \leq a$, then $a=b$. (Trainsitivity) If $a\leq b, \ b\leq c$, then $a \leq c...

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Random math: 06

Source of most content: Gunhee Cho Group Def. Given a set $S$, we call $* : S\times S \rightarrow S$ a binary operation if $* $ satisfies associatvie law, $\forall x,y,z \in S, \ (x * y) * z = x * (y * z) $, and we call $(S, *) $ the semi group. Def. Given a semi group $(M, *)$, if $\exists e \in M$ s.t. $\forall a \in M, \ a * e = e * a = a$...

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Random math: 05

Source of most content: Gunhee Cho MCT and NIP MCT Monotone Convergence Theorem (MCT) \[\text{If } (a_n)^\infty_{n=1} \text{ is a monotone increasing (decreasing) sequence, i.e.,} \forall n \in \mathbb{N}, \ a_n \leq (\geq) a_{n+1}. \nonumber\] \[\text{ Then, } \exists \lim a_n = \sup\{a_n \mid n \in \mathbb{N} \} \ (\inf \{a_n \mid n \in \...

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Random math: 04

Source of most content: Gunhee Cho Topology Topology is useful to classify geometric objects. Def. Given a set $X \ne \emptyset$, say $T \subseteq 2^X$ is a topology on $X$ if $T$ satisifes: $\emptyset, X \in T$ If $\forall \alpha \in I, \ E_\alpha \in T$, then $\cup_{\alpha \in I}E_\alpha \in T$ If $E_1, \cdots, E_n \in T$, th...

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Random math: 03

Source of most content: Gunhee Cho Density in $\mathbb{R}$ Density of $\mathbb{Q}$ \[\text{For any }x, y \in \mathbb{R} \text{ with }x < y,\text{ there exists }r \in \mathbb{Q}\text{ s.t. }x < r < y. \nonumber\] Proof It is suffice to consider the case $0 < x < y$. Since $y-x > 0$, by A.P. with 1, $\exists n \in \mathbb{N}$...

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Random math: 02

Source of most content: Gunhee Cho Least upper bound property This property is equivalent to completeness axiom in real space. Def. Let $E\subseteq \mathbb{R}$, say $E$ is bounded above (below) if there exists $\beta \in \mathbb{R}$ ($\alpha \in \mathbb{R}$) s.t. for each $x\in E$, $x \leq \beta$ ($x \geq \alpha$) In this case, $\beta$ is an ...

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Random math: 01

Source of most content: Gunhee Cho Function Def. Given two sets $A$ and $B$, we call $f: A \rightarrow B$ a function if we have $S_f \subseteq A \times B$ satisfying for each $a \in A$, there exists $b\in B$ uniquely s.t. $(a,b) = (a, f(a)) \in S_f \subseteq A \times B$. Here, $A$ is the domain, $B$ is codomain and $\text{Im}f = \{ b=f(a) | a \...

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