- Accounting Homework Help
- Biology Homework Help
- Chemistry Homework Help
- Computer Science Help
- Economics Homework Help
- Engineering Homework Help
- English Homework Help
- Essay Writing Services
- Finance Homework Help
- Management Homework Help
- Math Homework Help
- Matlab Programming Help
- Online Exam Help
- Online Quiz Help
- Physics Homework Help
- Statistics Homework Help

- Physics Assignment Help
- Chemistry Assignment Help
- Math Assignment Help
- Biology Assignment Help
- English Assignment Help
- Economics Assignment Help
- Finance Assignment Help
- Statistics Assignment Help
- Accounting Assignment Help
- Computers Assignment Help
- Engineering Assignment Help
- Management Assignment Help

Describe the fundamentals and basics on which mathematical theories are constructed

Mathematical theories are constructed starting with some fundamental assumptions, called axioms, such as "sets exist" and "objects belong to a set" in the case of naive set theory, then proceeding to defining concepts(definitions) such as "equality of sets", and "subset", and establishing their properties and relationships between them in the form of theorems such as "Two sets are equal if and only if each is a subset of the other", which in turn causes introduction of new concepts and establishment of their properties and relationships. Proofs are the arguments for establishing those properties and relationships. At the bottom level these arguments follow the inference rules of propositional and predicate logic, that is the conclusion of the theorem being proved must be derived from its hypotheses, axioms, definitions, and proven theorems using inference rules. However, at the bottom level they become tedious and inefficient as one can easily imagine. Thus in actual proofs short-cuts are taken using already proven theorems, using multiple inference rules in one step without explicitly mentioning them individually, omitting "obvious" proofs, and so on.