Note on Introduction to Mathematical Reasoning
Axioms 3.1.2
1. Trichotomy law: For each pair of real numbers a and b, one and only one of the three possibilities is true,
$ a\lt b \lor a = b \lor a\gt b.
2. Addition law: For real numbers a, b, and c,
$ a \lt b \leftrightarrow a + c \lt b + c.
3. Multiplication law: For real numbers a, b, and c,
$ a \lt b \leftrightarrow ac \lt bcif$ c \gt 0
$ a \lt b \leftrightarrow ac \gt bcif$ c \lt 0.
4. Transitive law: For real number a, b, and c,
$ a \lt b \land b \lt c \rightarrow a \lt c.
Definition 10.1.1
Given a set X, if there is a bijection$ f: \mathbb{N}_n \rightarrow X, then we say that the cardinality of X, or the number of elements in X, is n and write$ |X| = n. The cardinality of the empty set is$ |\emptyset| = 0.
* $ \mathbb{N}_n = \{1, 2,..., n\} = \{i\in \mathbb{Z}\mid 1\le i \le n \}