集合部分：
\(if \ A \subseteq B, \ B \subseteq A, \ then \ A = B\)
\((\cup_{\alpha \in A} S_{\alpha})^c = \cap_{\alpha \in A} (S_{a})^c\)
有序集部分：
\(if \ < is \ an \ order \ on \ S, \ these \ two \ followings \ hold:\)
\(1. \forall x,y \in S, \ x = y \ or \ x < y \ or \ y < x\)
\(2. if \ x < y, \ y < z, \ x < z\)
Suppose \((S, <)\) is an ordered set and has the least upper bound property. Then for any nonempty \(B \subseteq S\), which is bounded below, it has the greatest lower bound in \(S\).
Moreover, if \(L\) is the set of lower bounds of \(B\) in \(S\), then \(\sup L\) exists, and \(\sup L = \inf B\)
实数域与实数部分：
Archimedean property：
Suppose \(x,y \in R\) and \(x > 0\), there is some integer \(n\) such that \(nx > y\)
Denseness of \(\mathbb{Q}\) in \(\mathbb{R}\):
Suppose that \(x,y \in R\), and \(x < y\), there exists some real numbers \(z\) that \(x < z < y\)
Corollary:
Suppose \(x,y \in R\) and \(\forall \epsilon \in \mathbb{Q}\) and \(\epsilon > 0\), if there are \(a_{\epsilon}, b_{\epsilon}\) such that \(x,y \in [a_{\epsilon}, b_{\epsilon}]\) and \(b_{\epsilon} - a_{\epsilon} < \epsilon\) , then \(x = y\).