Mathematical Systems

Consists of the following:

1. A set of elements (for Real Number System, elements are all real numbers)
2. One or more relations expressing some kind of comparison among elements in the set ($<,>,=,\ge ,\le$)
3. One or more operations on the set of elements ($+,-,\ast ,/$, exponents, radicals)
4. A list of basic rules or laws that govern the behavior of the elements under the operations and relations (Axioms)

Examples: Real Number System ($\mathbb{R}=x\in -\infty <\infty$), set theory (chapter 3), binary systems (only elements are $1$ and $0$).